Wikiversity:Colloquium

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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:15, 9 May 2026 (UTC)}}
How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
{{archive bottom}}

== Wikiversity:Curators to become a policy ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:16, 9 May 2026 (UTC)}}
I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)
{{Archive bottom}}

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
{{archive bottom}}

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== Create a pseudo-bot user group? ==
{{tracked|T426882}}
I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)
: I'll file a Phabricator task by tomorrow if there are no objections. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:01, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 May 2026 (UTC)

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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      <id>2810762</id>
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        <id>10242</id>
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      <comment>/* Curators and curators policy */ archive to [[Wikiversity:Colloquium/archives/October 2025#Curators and curators policy]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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&lt;!-- MESSAGES GO BELOW --&gt;
== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== Wikiversity:Curators to become a policy ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:16, 9 May 2026 (UTC)}}
I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)
{{Archive bottom}}

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
{{archive bottom}}

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== Create a pseudo-bot user group? ==
{{tracked|T426882}}
I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)
: I'll file a Phabricator task by tomorrow if there are no objections. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:01, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 May 2026 (UTC)

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
{{archive bottom}}

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== Create a pseudo-bot user group? ==
{{tracked|T426882}}
I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)
: I'll file a Phabricator task by tomorrow if there are no objections. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:01, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 May 2026 (UTC)

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
{{archive bottom}}

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== Create a pseudo-bot user group? ==
{{tracked|T426882}}
I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)
: I'll file a Phabricator task by tomorrow if there are no objections. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:01, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 May 2026 (UTC)

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)
::::A matching accound makes sense to give credits to the original authors and keep a clean chain of versions. The initial commit into wikiversity could have a &quot;marker with timestamp&quot; similar to signature with a reference where the content's source or a Web archive. This would allow authors to continue there work on wikiversity if they wish. [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:30, 15 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)
: This was completed on 30 April.
: Perhaps we could benefit from some documentation (e.g., [[Wikiversity:Patrol]] or [[Wikiversity:Patrolling]]?) and updates to the curator, custodian, bureaucrat pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:49, 19 May 2026 (UTC)
:: Yes, but I would recommend [[Wikiversity:Patrolling]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:26, 19 May 2026 (UTC)

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Reminder about custodian-related pages ==

I would like to remind the community about what the following custodian pages are:
* [[Wikiversity:Request custodian action]] is for requesting actions to be done by custodians, and
* [[Wikiversity:Notices for custodians]] is for notices of interest to custodians, like an administrator's noticeboard
Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:12, 12 May 2026 (UTC)

:Thanks - I needed this reminder :) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:21, 12 May 2026 (UTC)

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== Create a pseudo-bot user group? ==
{{tracked|T426882}}
I would like to propose adding a new user group to Wikiversity: Pseudo-bot (&lt;code&gt;flood&lt;/code&gt;). This will allow users to perform repetitive actions without flushing the recent changes feed (with only the &lt;code&gt;bot&lt;/code&gt; user right). However, I would suggest that for the pseudo-bot user group:
* It can be granted and revoked by custodians. &lt;s&gt;However, can curators add and remove pseudo-bot from their own accounts (and not others)?&lt;/s&gt;
* Users can remove themselves from it.
* A guideline might be necessary about the information and usage of it.
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:31, 14 May 2026 (UTC)

:This sounds good. Which other wiki could we model this user group on? e.g., [[b:Wikibooks:Pseudo-bots]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:19, 15 May 2026 (UTC)
::@[[User:Jtneill|Jtneill]] Wikiquote has a similar group: [[:wikiquote:Special:ListGroupRights]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:25, 15 May 2026 (UTC)
: Should we allow curators to add and remove themselves from the pseudobot user group (from their own account) as well? I see no objections to creating the user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 18 May 2026 (UTC)
::My thinking is perhaps not curators by default because there should be clear visibility about their actions until they are well trusted. Let's draft a guideline or proposed policy ([[Wikiversity:Pseudo-bots]]) for the proposed user group. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:39, 18 May 2026 (UTC)
::: A solution is that they can ask any custodian to grant that group, and to remove themselves when done. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:17, 19 May 2026 (UTC)
:::: Yes, that sounds good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:12, 19 May 2026 (UTC)
: I'll file a Phabricator task by tomorrow if there are no objections. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:01, 19 May 2026 (UTC)

== Coming over From wikinews ==
Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)
::::::im ready to write @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 21:30, 13 May 2026 (UTC)
:::::::I think we should get more local consensus for a big project like including the entirety of the scope of Wikinews here. Again, I support it personally. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:55, 13 May 2026 (UTC)
::::::::ok lets begin. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:15, 13 May 2026 (UTC)

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)
:::Wondering, should we also have:
:::* {{tl|Inactive custodian}}
:::* {{tl|Inactive bureaucrat}}
:::or perhaps just a single template with a parameter(s) for the user right(s)/role(s)? e.g., 
:::* if a custodian is inactive for 2 years, then custodian and curator rights are to be removed and 
:::* if a bureaucrat is inactive for 2 years, then bureaucrat, custodian, and curator rights are to to be removed.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:58, 13 May 2026 (UTC)
:::: I would probably modify that template when we actually develop our own inactivity policy, because we're currently under the AAR (a steward notifies the colloquium with [[m:Admin activity review/Notice to communities]], and inactive advanced right holders with [[m:Admin activity review/Notice to inactive right holders]]). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 13 May 2026 (UTC)
:::::Ah, I see. Yes, that makes sense. Thankyou. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:21, 15 May 2026 (UTC)
: In that case, should we develop our own inactivity policy (e.g. on [[Wikiversity:Inactivity policy]] or [[Wikiversity:Support staff/Inactivity]])? I would list the general inactivity part, the process, etc. Once it's approved as a policy, I will [[m:Stewards' noticeboard|notify the stewards]]. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:30, 16 May 2026 (UTC)
::Originally, I would have thought that, for a small wiki like en.wv, it made sense to leave inactivity monitoring to the stewards. However, with the creation of the curator user group, we have already taken on local responsibility for monitoring inactivity in at least one advanced-rights group. Extending this to custodians and bureaucrats would not add much additional overhead and would provide a more consistent and transparent local administrative process.
::One option would be to develop a single, centralised policy covering all advanced-rights groups.
::An alternative would be to include an ==Inactivity== section on each relevant policy page (e.g., we already have [[Wikiversity:Curatorship#Inactivity]], but not yet in the custodianship, and bureaucratship policy pages). This approach would allow some flexibility because different user groups may warrant different criteria (such as inactivity thresholds, qualifying activity, or review procedures).
::A hybrid approach may be best: maintain separate inactivity sections within each user-group policy page, while transcluding these into a central overview page such as Codename Noreste suggests. This would preserve clarity at the local policy level while also providing a single reference point for consistency and oversight. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:09, 16 May 2026 (UTC)
::: I would suggest we develop a centralized inactivity policy page, and include a short summarized section of that page, on the support staff user group pages. We must also include a link to that policy page if we were to add &lt;nowiki&gt;== Inactivity ==&lt;/nowiki&gt; to each of those user group pages. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:48, 17 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

==Curator inactivity review==
These curators haven't been active for &gt; 2 years. As per the [[Wikiversity:Curatorship|curatorship policy]]:
* [[Special:Log/Cody naccarato]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Praxidicae]] was notified on their talk page by [[User:Codename Noreste|Codename Noreste]] on 24 Apr 2026
* [[Special:Log/Tegel]] was notified on their talk page by [[User:Jtneill|Jtneill]] notified their talk page on 16 May 2026

The policy allows a month to hear from these users. If no response, a custodian will remove their curator rights.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:14, 16 May 2026 (UTC)
: For Cody naccarato and Praxidicae, their rights are to be removed by the 19th of May if they don't respond either here or on their talk page. For Tegel, the removal will happen on the 16th of June, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:13, 16 May 2026 (UTC)
::Should be 24 May for Cody naccarato and Praxidicae? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:11, 16 May 2026 (UTC)
::: I made [[#Inactive curators]] on the 19th of April. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:18, 17 May 2026 (UTC)
::::OK, I see (had missed that thread, sorry - I've now moved the the 3 inactivity topics to be adjacent).
::::I'm thinking the curator policy indicates one month from user talk page notification? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:44, 17 May 2026 (UTC)
::::: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:49, 17 May 2026 (UTC)
: @[[User:Juandev|Juandev]] and @[[User:Jtneill|Jtneill]]: feel free to remove Cody naccarato and Praxidicae's curator permissions. They have not responded at all after one month. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:29, 19 May 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==

[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===</text>
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== Call for custodians and bureaucrats ==
&lt;div class=&quot;cd-moveMark&quot;&gt;''Moved from [[Wikiversity:Request custodian action#Call for custodians and bureaucrats]].  [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:46, 12 May 2026 (UTC)''&lt;/div&gt;

Can I encourage currently active [[Wikiversity:Curators|curators]] to consider putting themselves forward for [[Wikiversity:Custodianship|custodianship]] and/or [[Wikiversity:Bureaucrat|bureaucratship]]. We have a productive, capable group of [[Wikiversity:Staff|staff]] at the moment who should probably have more rights to better support the project and we are light on for active custodians and bureaucrats. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:34, 9 May 2026 (UTC)
: I'm willing to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 11:48, 9 May 2026 (UTC)
::Awesome. Could you self-nominate at [[Wikiversity:Candidates for Custodianship]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:59, 11 May 2026 (UTC)
::: I filed my nomination, but according to the custodianship policy, I am running for probationary custodianship, and after a period of four weeks, I will run again for permanent custodianship to determine if I have performed well and professionally. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:00, 12 May 2026 (UTC)
:I'm also willing to run for bureaucratship as I imagine my activity levels should remain sufficient. I could put in a nomination within the next week or so. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:55, 11 May 2026 (UTC)
::Wonderful. [[Wikiversity:Candidates for Bureaucratship]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:09, 12 May 2026 (UTC)
:Would also like to help! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:25, 13 May 2026 (UTC)
::Merci beaucoup :) When you're ready, you can self-nominate for probationary custodianship at [[Wikiversity:Candidates for Custodianship]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:08, 14 May 2026 (UTC)
: Would an uninvolved bureaucrat close the following discussions? Roughly a week has passed. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 20 May 2026 (UTC)

== Call for custodian mentors==
If you have more than 3 months experience as a custodian, please consider listing yourself as potential mentor for probationary custodians: [[Wikiversity:List of custodian mentors]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:27, 12 May 2026 (UTC)</text>
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== Call for custodians and bureaucrats ==
&lt;div class=&quot;cd-moveMark&quot;&gt;''Moved from [[Wikiversity:Request custodian action#Call for custodians and bureaucrats]].  [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:46, 12 May 2026 (UTC)''&lt;/div&gt;

Can I encourage currently active [[Wikiversity:Curators|curators]] to consider putting themselves forward for [[Wikiversity:Custodianship|custodianship]] and/or [[Wikiversity:Bureaucrat|bureaucratship]]. We have a productive, capable group of [[Wikiversity:Staff|staff]] at the moment who should probably have more rights to better support the project and we are light on for active custodians and bureaucrats. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:34, 9 May 2026 (UTC)
: I'm willing to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 11:48, 9 May 2026 (UTC)
::Awesome. Could you self-nominate at [[Wikiversity:Candidates for Custodianship]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:59, 11 May 2026 (UTC)
::: I filed my nomination, but according to the custodianship policy, I am running for probationary custodianship, and after a period of four weeks, I will run again for permanent custodianship to determine if I have performed well and professionally. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:00, 12 May 2026 (UTC)
:I'm also willing to run for bureaucratship as I imagine my activity levels should remain sufficient. I could put in a nomination within the next week or so. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:55, 11 May 2026 (UTC)
::Wonderful. [[Wikiversity:Candidates for Bureaucratship]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:09, 12 May 2026 (UTC)
:Would also like to help! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:25, 13 May 2026 (UTC)
::Merci beaucoup :) When you're ready, you can self-nominate for probationary custodianship at [[Wikiversity:Candidates for Custodianship]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:08, 14 May 2026 (UTC)
: Would an uninvolved bureaucrat close the following discussions? Roughly a week has passed. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 20 May 2026 (UTC)
:: Ping @[[User:Guy vandegrift|Guy vandegrift]] @[[User:Dave Braunschweig|Dave Braunschweig]] @[[User:Mu301|Mu301]]: Two probationary custodian nominations are ready for closing if you're available. Also note that there are two bureaucrat nominations that should stay open for another week or so. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:57, 21 May 2026 (UTC)

== Call for custodian mentors==
If you have more than 3 months experience as a custodian, please consider listing yourself as potential mentor for probationary custodians: [[Wikiversity:List of custodian mentors]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:27, 12 May 2026 (UTC)</text>
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== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
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== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
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== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
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== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
{{archive bottom}}

== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

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== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)

:'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Keep''' and integrate with existing [[:Category:Filmmaking]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 20 May 2026 (UTC)

==[[United States UFO files]]==
Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:46, 21 May 2026 (UTC)</text>
      <sha1>ldhvp1h9cf09x9e116lb2i683agmz2d</sha1>
    </revision>
    <revision>
      <id>2810714</id>
      <parentid>2810708</parentid>
      <timestamp>2026-05-21T02:08:46Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* United States UFO files */ Reply</comment>
      <origin>2810714</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="60644" sha1="apnk53g6iepq6i2k3uwfc6fzatlwv4i" xml:space="preserve">{{/header}}
== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
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== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
{{archive bottom}}

== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
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== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
{{archive bottom}}

== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

{{archive bottom}}

== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)

:'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Keep''' and integrate with existing [[:Category:Filmmaking]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 20 May 2026 (UTC)

==[[United States UFO files]]==
Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:46, 21 May 2026 (UTC)

:'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: &quot;Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.&quot;)? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: &quot;No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly.&quot;).
:Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC)</text>
      <sha1>apnk53g6iepq6i2k3uwfc6fzatlwv4i</sha1>
    </revision>
    <revision>
      <id>2810715</id>
      <parentid>2810714</parentid>
      <timestamp>2026-05-21T02:20:45Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* United States UFO files */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2810715</origin>
      <model>wikitext</model>
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      <text bytes="61168" sha1="rj7sav5e0xhpdra80zx7fwemll282ns" xml:space="preserve">{{/header}}
== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

I proposed to quasi-delete, i.e. '''move to userspace''' of the main (or sole?) creator, {{User|KYPark}}.

The page is organized a little bit like a dictionary. It makes it redundant to Wiktionary except that Wikiversity allows original research and there does seem to be original research there. Thus, its being organized as a dictionary would alone not necessarily be a problem.

Where I see a problem is in the organization and execution/implementation. Consider [[Korean/Words/가다]], which seems rather typical of the subpages (some subpages are like categories and transclude the pages for individual words):
* On the putative definition line, there is this: &quot;한곳에서 다른 곳으로 장소를 이동하다&quot;, apparently(?) in Korean. That does not seem to fit well into the ''English'' Wikiversity.
* There seems to be some original research into etymological relations between Korean and European languages in the &quot;Comparatives&quot; section (from what I recall, the English Wiktionary rejected this kind of content from KYPark). Admittedly, it is marked using &quot;This is a primary, secondary and/or original Eurasiatic research project at Wikiversity&quot;, so it could be tolerable, but even so, one has to wonder whether Wikiversity wants this kind of fringe science/research or outright pseudo-science.
** Fringe science: fringe physics has been moved to user space before. This would be fringe etymology. But then, original research is allowed.

Deletion is not required; moving to user space suffices, I think. Alternatively, one could at least rename the pages to make it clear from the title that this is not Wikiversity voice but rather KYPark voice, e.g. &quot;Korean/Words (KYPark)/...&quot; or  &quot;Korean/Words/KYPark/...&quot;  (recall the &quot;Fedosin&quot; pages featuring the name &quot;Fedosin&quot;).

Methodology: I see almost no methodological notes spanning the words at [[Korean/Words]]. And yet, if this is original research inventing new etymological connections, surely there should be some general considerations/analysis on how to proceed and how that manner of procedure differs from mainstream etymology?

Prefix index (max 200 items?):
{{collapse top}}
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}
{{Collapse bottom}}
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

:I would keep it. If there is a course of Korean, why not to have a resesearch on Korean vocabulary? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:53, 16 October 2025 (UTC)
:: I propose to dismiss the above input: 1) it does not contain any argument, except for a question, and a question is not an argument (it can be so reinterpreted, but that includes additional burden on the interpreters, in interpreting it the wrong way); 2) it ignores all the issues I have raised, including that there is something like definition lines in Korean, in this ''English'' Wikiversity. To answer the question asked: there can be a research on Korean vocabulary in the mainspace, but not one showing the defects I identified above. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:35, 15 November 2025 (UTC)
:I've reviewed a sample of approximately 20 of the Korean/Words sub-pages and lean towards moving to user space because:
:* The pages appear to be an idiosynchratic collection of etymological pages about Korean language
:* There is minimal English instruction which is problematic for English Wikiversity
:* There is no explanation of research method
:* There is no educational rationale
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:31, 22 November 2025 (UTC)
:Well, since the original creator has indef I change my mind and I would '''delete''' it. The case is nobody knows how to continue with the research and if we move it to the userspace, the user cannot improve it eihter. What the original user can do to request admin, to send them a contentent to their email for example if they really want to improve the resource elsewhere. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 11 March 2026 (UTC)

I think the consensus here is delete. {{U|Codename Noreste}} do you know an efficient way to mass delete these pages? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 17 May 2026 (UTC)
: I would use a script, but I would probably not delete those pages yet until we have the pseudobot user group. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:53, 17 May 2026 (UTC)

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
{{archive top|'''Kept'''. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 17 May 2026 (UTC)}}
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

I am not sure about how to proceed, although I am inclined to move it out of mainspace = quasi-delete. I am looking forward to get input from others, especially curators and custodians. Some considerations:

1) There is perhaps no more appearance/suspicion of copyright violation, now that the ResearchGate (RG) article (of which this is a copy, perhaps an incomplete copy?) carries a license.

2) The article is not a complete replica from RG: at a minimum, it lacks images. The inserter could have continued editing the page in his user space before he uploads images, that is, before he finalizes the page for consumption, but that did not happen. I did not check whether the text is an exact one-to-one match; the article does not indicate anything in that regard.

3) The principle implied seems to be this: users should feel free to duplicate non-peer-reviewed articles from RG in English Wikiversity, perhaps to increase the Google search and LLM yield. I find this problematic, in part for the duplication. I would say: choose a venue and publish it there. If RG is not good enough for you as a publishing venue, choose Wikiversity instead, but not both?

4) There are some features that appear unduly promotional. There is a link to a dot com home page of the inserter of the article. I dot not know how we handle or should handle this, whether prohibit such a link, etc. This is perhaps not so much a call to quasi-deletion but a call to make it less promotional.

5) I cannot determine the value of such an article. It seems to be a pseudo-article describing someone's browser extension. Can someone do a better analysis?

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:48, 8 October 2025 (UTC)

:2) Images for Wikicommons are being created, it will take a lot of time. and the text is not an exact one-to-one match
:3) I also mentioned that It was being created so that it is more accessible from mobile phone, which is not possible in RG or in Zenodo
:Let me clarify the purpose of uploading it to different platforms
:Zenodo - registration and to link DOI
:RG - Self Archiving 
:Wikiversity - Accessible by anyone from any device. LLMs may get trained on Wikiversity data or use these data for indexing 
:5) The paper is a result of a research project which involved a browser extension which was built to test the theory.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 01:34, 9 October 2025 (UTC)
:: I find the practice here of publishing non-identical but similar text (&quot;the text is not an exact one-to-one match&quot;) with almost the same title to be problematic. I cannot imagine this is a recommended practice in academic publishing. At a minimum, somewhere near the top, the page should say something like the following: &quot;This text is based on article ___ published at ___ but is not identical. The author of the differences/changes is ___.&quot; Everything else leads to an undesirable confusion. In academic publishing, the title of an article serves as key part of identification of the artifact.
:: As I said before, I seen nothing particularly academic article-like about the page except for external/superficial signs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:30, 9 October 2025 (UTC)
:::That Article has been published under CC BY SA 4.0
:::And I am one of the author of the article. That gives me right to modify text and publish it under a similar name. However, I will add the disclaimer text that you have suggested. I hope that helps.  [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:07, 9 October 2025 (UTC)
:::: It may give you that right from the ''copyright'' perspective, but perhaps not from ''academic publishing integrity'' perspective. Unfortunately, I do not have any guideline handy; I am merely following my common (or not so common) sense. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:32, 9 October 2025 (UTC)
:: I would like to ask: was this article guided by someone from an academic institution, such as a university? Is it reviewed at least in some weak sense? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:39, 9 October 2025 (UTC)
:::Yes, This article has been reviewed by two academic professors, their names are also listed as co authors.
:::First, a project guide would help us with selecting a topic and with the document 
:::Second, an Internal examiner would go through our experiment and approve it
:::Finally, External Examiner would examine the documentation and verify it. 
:::We were required by these professors to put their name under contributions [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 05:48, 9 October 2025 (UTC)
:: Let me explicate the promotional potential of such a page a bit: one can go to the page of the article in Wikiversity --&gt; https://tomjoejames.com/ --&gt; HitMyTarget (a commercial, profit-making entity?) Why would the link be to a commercial web site rather than an academic page, or perhaps a LinkedIn account, which I think the person has? There could also be no link at all; a search for the name would turn out something in Google as well. But providing a direct link would drive users/viewers toward that website much stronger since otherwise the viewer of the page would have to open a new Google search window or the like. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:45, 9 October 2025 (UTC)
:::It is evident that the website is not even close to being complete. 
:::I will be creating a separate page under the same domain name specifically for people to contact me.
:::The url would probably be defined as tomjoejames.com/contact-me/
:::I haven't decided yet. But that is my personal website.
:::If the community requires me to remove it, I will. But personally I think people who are from here most likely to click the link to know more about me or to contact me. Either way I think my personal website serves the purpose. 
:::As for the HitMyTarget, it can be traced from any of my links. From my research gate profile, linkedin page or even my own userpage.
:::On the article I did not add any promotional content about myself, I hyperlinked only my own name. I do not know how that is promotional. [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 06:04, 9 October 2025 (UTC)
:::: I am pausing any further responses from me to see whether anyone else has any input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:30, 9 October 2025 (UTC)
:What does it mean &quot;There is perhaps no more appearance/suspicion of copyright violation&quot;? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:57, 16 October 2025 (UTC)
:I have accepted VRT permission per [[ticket:2025100410001149]] FYI. [[User:Matrix|Matrix]] ([[User talk:Matrix|discuss]] • [[Special:Contributions/Matrix|contribs]]) 11:00, 28 October 2025 (UTC)
::Thank you Matrix [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 12:43, 28 October 2025 (UTC)
:I would '''delete''' it. 1) it states its a learning resource. It could not be a learning resource as not rewieved original research. 2) It is not an ongoing research, nor the research was performed on Wikiversity - wv is not a preprint or article database. Maybe it could be moved elsewhere withn Wikimedia domain, but I dont know where. So I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)
::I would '''keep it.''' Like Dan had pointed out,  we do have article-like pages in Wikiversity, and this is not just a random pseudo science article but an article that is a report of an final year project, it has been reviewed by 3 professors whose name has been mentioned at the very beginning.   [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:50, 21 November 2025 (UTC)
:::I think it is not good to rate pages by appearance. It can be done on other Wikimedia projects, but it cannot be done on Wikiversity, because Wikiversity does not create a static format for presenting information, but is focused on the goal and process. Unfortunately, the goal and process do not have a uniform format. While a target article on Wikipedia or an entry on Wiktionary have some standard target format, Wikiversity does not. That is why I personally rate pages according to the goals and their assessment. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:05, 22 November 2025 (UTC)

Further reading for this nomination: [[S: Wikisource:Proposed_deletions/Archives/2025#Index:Cookie_Encryption.pdf]]; EncycloPetey handled the matter. Let me quote his wisdom on Zenodo (which I lack): &quot;This is tied to a PDF on Commons that was uploaded as &quot;own work&quot; with a CC license and a doi link to Zenodo, with no indication of where this paper was published or if it was published. Zenodo is not a publisher; it is a site for storing research and sharing papers. If Zenodo is the only place this was &quot;published&quot; then it was effectively self-published. --EncycloPetey (talk) 16:14, 15 September 2025 (UTC)&quot;

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:55, 9 October 2025 (UTC)

:Can you clarify what point are you trying to state? Didn't I already state that the article is published by me? 
:I first created the article in wikisource which I thought would be the perfect place, unfortunately they do not allow self published articles that are not notable. Then I discovered Wikiversity where they allow self published articles. That is why I created the article here. 
:Unlike in wikisource, I did follow guidelines.
:Ever since you deleted the first article, I spent time reading Wikiversity guidelines and I do think that I am following it perfectly.
:I would like to get your suggestions on how should I improve the page,  10 points would be sufficient. 
:Because your goals or intentions are confusing me very much. At first you told me that the article is exactly the same as the preprint in RG and therefore there is no use to it here. And then when I continued to optimize it for Wikiversity, you went ahead and said it is problematic according to recommended academic publishing. 
:Atleast just respond to the points that I have made whether you agree or disagree. So that I clarify and proceed to discuss points that are important and relevant 
:Have you published an research article? If yes, could you send it to me so that I can see the format you have done it [[Special:Contributions/&amp;#126;2025-27520-79|&amp;#126;2025-27520-79]] ([[User talk:&amp;#126;2025-27520-79|talk]]) 10:45, 9 October 2025 (UTC)
:: I am giving a chance/time to other curators/custodians to look at the matter and respond to my inputs. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:14, 9 October 2025 (UTC)
:: Incidentally, above I counted 4 questions (or more), 1 request (or more?) and 1 command (or more?). That is a behavior of a commanding entity. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 9 October 2025 (UTC)

I would '''delete it''''. It's more like an academic communication than a learning resource or research.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:32, 26 October 2025 (UTC)
:: In the above post, I do not see any valid rationale for deletion: we do have article-like pages, in Wikijournals and also e.g. in [[Physics/Essays/Fedosin/Stellar Stefan–Boltzmann constant]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:59, 3 November 2025 (UTC)
:::But I do, see above. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 20 November 2025 (UTC)

:it is a '''student research paper''' forming part of a learning resource on web security and encryption.
:The project was conducted as part of a final-year university course and documented as a practical study on cookie encryption and it has been reviewed by three professors. However, I will be creating a sub page for the article to elaborately describe the experiment that we have conducted and the results we got.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 15:57, 26 October 2025 (UTC)
::And why should w host research papers? Wikiversity is not an academic Journal nor repository. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:06, 22 November 2025 (UTC)
:::I do not wish to go through this same argument once again, I've already answered to this question several times in Dan's talk page, Colloquium. you can refer them. I am not hosting the research paper here, I have already hosted the pdf in the ResearchGate, I have published a text version in the wikiversity so that it may be useful for others. Unless you can show me how that article is totally useless, I would like to '''keep''' the article in the wikiversity.    [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 10:13, 22 November 2025 (UTC)
::::And thats the point I am having. Wikiversity is not paper repository. The only way is to publish it via WikiJournal, but they want it for Wikipedia usually. Why wikiversity should be a duplication of ResearchGate, Academia or Zenodo?
::::What I can read on [[Wikiversity:What is Wikiversity?]] policy is, that Wikiversity research &quot;...includes interpreting primary sources, forming ideas, or taking observations.&quot; The article doent look to fall into this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:43, 22 November 2025 (UTC)
:::::Well, then how come you missed the term &quot;Learning Projects&quot;? As Jtneill had pointed out, this is a legitimate learning project. And also, I do have the VRT permission to host this article on Wikiversity.  [[ticket:2025100410001149]] . besides ResearchGate is an self-archiving platform. the document version in it is not accessibly to screen readers (usually disable people use them), Translators, and also for the mobile readers. therefore I do have valid reasons to publish this article on wikiversity. 
:::::# It is a learning project, therefore according to WIkiversity Policy, It qualifies. 
:::::# I have an explicit VRT permission to host this article on Wikiversity
:::::# Versions that are published in RG, Zenodo are documents, and they are not accessible by screen readers or mobile users. Therefore it is imperative that an article version of this paper exist on here. 
:::::Therefore this article qualifies to stay here on Wikiversity.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 11:22, 22 November 2025 (UTC)

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)
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== [[Pragmatics/History]] ==
{{archive top|Deleted. Other related resources have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:24, 17 May 2026 (UTC)}}
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

As an alternative, moving to [[History of Pragmatics (KYPark)]] would make sense to me: the topic is identified using a natural-language phrase (instead of the relatively unnatural slash) and the responsible editor is indicated so that the reader knows whether to look or not. And for those who oppose the brackets (which I like): [[History of Pragmatics/KYPark]]. Or also: [[KYPark/History of Pragmatics]]. But then, searches in mainspace will see that content and the question is whether that is good. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 15 October 2025 (UTC)

:What about to propose the user to write some guidelines, how other can participate instead of deleting it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:03, 16 October 2025 (UTC)
:: I plan to move the pages to userspace as I proposed. If someone wants to ask KYPark to address the problems, they should feel free. There will be plenty of time for KYPark to address the problems while the material is in user space. After the problems are addressed, the material can be moved back to mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:38, 15 November 2025 (UTC)
:So I would '''delete''' it. In the blocked user space its useless. The user cannot improve it and Wikiversity is not free hosting service for personal pages. My believe is, that there should be just a few working pages in the users spaces. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:30, 11 March 2026 (UTC)

'''Move'''. Insufficient statement of learning objective or connection to related learning resources with insufficient current activity to stay in main space. The page was originally [[History of pragmatics]] but was moved by Dave B. Therefore, I suggest moving to [[User:KYPark/History of pragmatics]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 22 November 2025 (UTC)
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== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)

== [[Palliative medicine]] ==
{{archive top|'''Kept'''. Page has been improved. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:38, 17 May 2026 (UTC)}}

Underdeveloped and has not been improved on since 2007. Author inactive. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:42, 14 December 2025 (UTC)

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)
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== [[Canadian Wilderness]] ==
{{archive top|Deleted per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:07, 17 May 2026 (UTC)}}
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

:In principle there could be some material useful here but in practice, I don't see what this page is adding as an educational resource. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:54, 6 February 2026 (UTC)
:I can see this being a useful resource to a bigger project. Maybe we could move it to the &quot;[[Wikiversity:Drafts|Draft]]&quot; namespace vs. deleting it? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:28, 6 February 2026 (UTC)
::Does anyone plan to work on it? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 01:59, 8 February 2026 (UTC)
:::Next week the page has it's 17th birthday. Ever now and than someone added to it. With a lot of work it could be a nice encyclopedic article but making it educational .... Merging it may take more work than rewriting it. Move to Draft might be the best option. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:58, 12 February 2026 (UTC)
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== [[LQR Control for an Inverted Pendulum]] ==
{{archive top|'''Deleted''' per nom. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:36, 17 May 2026 (UTC)}}
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

:Looks like a test, '''delete'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:30, 27 March 2026 (UTC)
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== False flag &quot;authority hack&quot; user page deletion ==
{{archive top|'''Not undeleted''', the requester dropped the request. See Wikiversity:Requests for Deletion v. 2803217.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 10 May 2026 (UTC)}}
'''Undeletion requested'''

Hi, Juandev marked my user page as &quot;spam&quot; and &quot;authority hack&quot;, and deleted it. 

First, I asked him for help with &quot;time limit for new users&quot;, and he replied - I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. 

Then he wrote me another message: Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. - probably referring to the intro of my About me page where I present me and my work. 

Before I could explain him the difference between the neutral information and advertising and promotion, he deleted my user page. 

Here is my answer I posted to the discussion today: 

: Hi, my About Me page is just an info page with the neutral as possible presentation of my work.
:
: There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that.
:
: There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
:
: Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies.
:
: Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):

: == Introduction ==
: The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
:
: The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:
: == Theoretical foundations ==
: The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
:* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
:* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
:* Narrative identity and partial‑self models within personality and identity theory.
: Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:
: == Experiential empiricism ==
: The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
:* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
:* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
:* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
:* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
:
: All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
:
: I believe this is a valid contribution to Wikiversity.
:
: Best regards, Senad [[User:Senad Dizdarević|Senad Dizdarević]]


I suggest you check the deleted user page, and see for yourself if it is &quot;spam&quot; and &quot;authority hack&quot;, or a legit author's page with one paragraph short presentation, while the rest of the page is about my research project. 

Thank you for undeleting my user page, so I can use it. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:26, 2 April 2026 (UTC)

:Hi Senad,
:Welcome to Wikiversity.
:It looks like you tried adding similar content to Wikipedia and ran into similar difficulties over there ([[w:User talk:Senad Dizdarević]])? Perhaps that is what has led to you Wikiversity?
:Basically, if you'd like to collaborate and help build open educational resources, feel free to contribute to Wikiversity. But if the primary motivation is to promote your autobiographical work you're probably going to run into challenges.
:Sincerely,
:James
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:11, 3 April 2026 (UTC)
::James, Hi, and thank you for your answer. 
::Yes, in 2025, I created the autobiographic page on Wikipedia, which was removed because of the links to my books on Amazon. To admin, I explained that I did not know the rules, and agreed that page is removed. Now I know that somebody else must write a Wikipedia page for you. 
::On the deleted user page on Wikiversity, there were no links to Amazon or any other form of promotion, just neutral as possible basic presentation of my writing (one sentence) and current project (the rest of the page). 
::I created Wikiversity page to present my AIPA Method project, to invite researchers to read it, give their opinion, and conduct empirical researches in their institutions. Now, it is in a theoretical phase, and needs more empirical testing. 
::Best regards, 
::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:03, 3 April 2026 (UTC)
:::It looks to me like the primary motivation for contributing to Wikiversity is to drive traffic / search engine ranking to your website?
:::* [[User:Senad Dizdarević]]
:::* [[AIPA Method]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:36, 4 April 2026 (UTC)
::::No, it is not. There is no link to my website, so &quot;driving traffic to my website&quot; is not possible. 
::::For your educational purposes: 
::::Copilot &quot;AI:  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:38, 4 April 2026 (UTC)
:::::So do you still insist of undeleting your former version of your userpage if you have created the new one? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:15, 6 April 2026 (UTC)
::::::No, because in the moment of undeletition, somebody could delete it again, and so on. Thank you for not deleting my new user page, as it is made in your user page image.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 08:59, 6 April 2026 (UTC)

{{archive bottom}}

== Undeletion request ==

It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)

==Pages by Harold Foppele==
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)

== [[Film writing]] ==

Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)

:'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Keep''' and integrate with existing [[:Category:Filmmaking]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 20 May 2026 (UTC)

==[[United States UFO files]]==
Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:46, 21 May 2026 (UTC)

:'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: &quot;Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.&quot;)? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: &quot;No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly.&quot;).
:Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC)
: Yes, could be speedy deleted. Otherwise, I don't know about the merits about leaving it around for 90 days, hence me bringing it to here. There is some comment in [[Wikiversity:Deletion policy]] about the specific deletion templates not being so important. More important I think is to flag for discussion. However, we could also improve the proposed policy to make the process clearer. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 21 May 2026 (UTC)</text>
      <sha1>rj7sav5e0xhpdra80zx7fwemll282ns</sha1>
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      <contributor>
        <username>Jtneill</username>
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      <comment>curators and custodians -&gt; support staff</comment>
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|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the page. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.
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&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. {{{notes|}}}''{{subpagesif}}&lt;/span&gt;
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    <title>Wikiversity:List of custodian mentors</title>
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      <comment>/* See also */ * [[Wikiversity:Support staff]]</comment>
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      <text bytes="1609" sha1="itkt6a14wgcf374pba6gngz62uk3l4y" xml:space="preserve">{{Administering Wikiversity}}
Candidate Wikiversity [[Wikiversity:Custodianship#How does one become a custodian?|custodians]] should select a mentor. Please make sure that the mentor you select is willing to mentor you. To make that completely clear, for each pair of a mentor and a custodian candidate, each must accept the other.

==List of experienced custodians who are available for mentoring new custodians==
&quot;'''Experienced custodians'''&quot; means Wikiversity custodians who have at least 3 months of experience as a sysop (&quot;administrator&quot; = &quot;custodian&quot;) on Wikimedia Foundation projects.

This list is in approximate order of recent activity. Names lower down on the list may be custodians who have mentored in the past but who are are not currently very active at Wikiversity.

*{{custodian|Dave Braunschweig}}
*{{custodian|Jtneill}}
*{{custodian|Mu301}}
&lt;!-- Not currently active *{{custodian|SB_Johnny}}

**Notes: Custodial tools are useful when held by trusted and dedicated contributors, even if they are rarely used, and just like editing and creating content the more people there are to share the work, the easier it is to get done what needs doing. Candidates should not feel obligated to become a full-time staff member, but rather should make sure they have time during mentorship period to learn how and when to use them. --&gt;

==See also==
* [[Wikiversity:Candidates for Custodianship]]
* [[Wikiversity:Custodianship]]
* [[Wikiversity:Support staff]]
* [[How to be a Wikimedia sysop]] - a project which helps people become sysops on Wikimedia projects

[[Category:Wikiversity administration]]</text>
      <sha1>itkt6a14wgcf374pba6gngz62uk3l4y</sha1>
    </revision>
    <revision>
      <id>2810773</id>
      <parentid>2810771</parentid>
      <timestamp>2026-05-21T11:33:38Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* List of experienced custodians who are available for mentoring new custodians */ +</comment>
      <origin>2810773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1631" sha1="bmqcwlsz4q3a9lp5msn6bzgzz6sox3j" xml:space="preserve">{{Administering Wikiversity}}
Candidate Wikiversity [[Wikiversity:Custodianship#How does one become a custodian?|custodians]] should select a mentor. Please make sure that the mentor you select is willing to mentor you. To make that completely clear, for each pair of a mentor and a custodian candidate, each must accept the other.

==List of experienced custodians who are available for mentoring new custodians==
&quot;'''Experienced custodians'''&quot; means Wikiversity custodians who have at least 3 months of experience as a sysop (&quot;administrator&quot; = &quot;custodian&quot;) on Wikimedia Foundation projects.

This list is in approximate order of recent activity. Names lower down on the list may be custodians who have mentored in the past but who are are not currently very active at Wikiversity.

*{{custodian|Atcovi}}
*{{custodian|Dave Braunschweig}}
*{{custodian|Jtneill}}
*{{custodian|Mu301}}
&lt;!-- Not currently active *{{custodian|SB_Johnny}}

**Notes: Custodial tools are useful when held by trusted and dedicated contributors, even if they are rarely used, and just like editing and creating content the more people there are to share the work, the easier it is to get done what needs doing. Candidates should not feel obligated to become a full-time staff member, but rather should make sure they have time during mentorship period to learn how and when to use them. --&gt;

==See also==
* [[Wikiversity:Candidates for Custodianship]]
* [[Wikiversity:Custodianship]]
* [[Wikiversity:Support staff]]
* [[How to be a Wikimedia sysop]] - a project which helps people become sysops on Wikimedia projects

[[Category:Wikiversity administration]]</text>
      <sha1>bmqcwlsz4q3a9lp5msn6bzgzz6sox3j</sha1>
    </revision>
  </page>
  <page>
    <title>Portal:Music/Introduction</title>
    <ns>102</ns>
    <id>4704</id>
    <revision>
      <id>2810647</id>
      <parentid>2769783</parentid>
      <timestamp>2026-05-20T19:27:58Z</timestamp>
      <contributor>
        <username>Kirby - Electrotechnics</username>
        <id>3074947</id>
      </contributor>
      <minor />
      <comment>Edited Active and Inactive Participants, and Guitar Tuner Resource to a separate column for review, due to the fact linking to the other site could be malware/an unhelpful recommendation</comment>
      <origin>2810647</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8948" sha1="14u0337gfi0af4t37x1odnwoyhd3dtu" xml:space="preserve">[[File:Musical notes.svg|right|120x120px]]

Music is a self-expressed art form whose medium consists of pulse, movement, tempo, pitch (measured by frequency and music theory scales), plucks (either on strings or percussion), friction (e.g, of strings) ambience (either performed live or in studio - e.g reverberation and absorption), vibration of sound waves, radio and electromagnetic alteration (such as noise and theremin), controlled-voltage electronic synthesis, collage/alteration of sound recordings (popularly known as samples) and predetermined moments of silence. 

	Whether an audible work is recognized as music it depends on the cultural context which is experienced; common agreement among participants (as this self-expressed art form can expand collectively, through a band/group/orchestra/collective, mantras and even through the listeners). 

	Despite the existence of atonal music, the predominant characteristics of this art are determined by melody - a more understandable method of frequency separation (into musical notes) - and harmony, which in most cases occur when non-frequency colliding sound waves are emitted simultaneously (except in cases of intentional dissonance). 

	Fundamentals related to pulse belong to rhythm (associated to the concepts of tempo, meter and articulation) that are determined by percussive instruments, drums and sometimes electrical pulses&lt;ref&gt;¹ Omri Cohen (2021-09-27) &quot;Audio-rate everything!&quot;https://www.youtube.com/watch?v=RxHHJIDQC0A&lt;/ref&gt;. Dynamics relates to overall loud and soft passages exerted by the instruments (not mentioning “compressors”&lt;ref&gt;² &quot;Three tricks with the FET Compressor. Softube Studios (2012)https://www.youtube.com/watch?v=sDMBdR1OW38&lt;/ref&gt;), just as the sonic qualities of timbre and texture of each (sometimes subjectively referred to as the “color” of a musical sound).

==Theory and composition==

===Western music===

The goal of the 'Theory and composition' department is to equip the student with the tools and skills necessary to compose, arrange and analyze music. At the completion of this course of study, students will possess the skills and knowledge of western theory, creative writing, arranging, as well as having a portfolio of original works.

{{colbegin|3}}
* [[Introduction to music|Introduction to music]]
* [[Fundamentals of Music|Fundamentals of music]]
* [[Music Theory I]]
* [[Music Theory II]]
* [[Chords (music)|Chords]]
* [[Harmony]]
* [[Form and Analysis|Form and analysis]]
* [[Beginning Composition|Composition : Beginning]]
* [[Advanced Composition|Composition, Advanced]]
* [[Lyrical Composition|Lyrical composition]]
* [[Counterpoint]]
* [[Music Technology]]
* [[Arranging]]
* [[Orchestration]]
* [[Film scoring for Musicians|Film scoring]] (in conjunction with the [[Course:Practical narrative film editing|Film editing course]])
* [[Final Theory Project]]
* [[Music Appreciation|Music appreciation]]
* [[Glossary of musical terms]]
* [[Xenharmonic music theory]]
{{colend}}

=== Non-western music ===
Some Non-European cultures have different music composition, arrangement and analysis traditions, less commonly known in western cultural spheres.{{colbegin|3}}
* [[Gamelan]]
* [[Carnatic music]]
* [[Andalusi Classical Music]]
* [[Persian Classical Music]]
* [[Arabic Classical Music]]
* [[Ottoman Classical Music]]
* [[Hindustani Classical Music]]{{colend}}
===Genres===
Some genres of Western music have genre-specific music theory.{{colbegin|3}}
* [[Basic Blues &amp; Rock]]
* [[Jazz]]{{colend}}


===Ear training===
Ear training is learning/training your ears to recognize what you hear and put it down onto paper. These are basic learning guides, exercises and projects to help you understand in a meaningful way the flurry of sound in music.{{colbegin|3}}
* [[Ear training - Intervals and Harmony|Ear training - Intervals and Harmony (pitch oriented)]]
* [[Ear training - rhythm]]{{colend}}

==Musicology==

{{MultiCol}}
=== Generalities ===

* [[Music Appreciation|Music appreciation and history]]
* [[Survey of Musical Genres|Survey of musical genres]]
* [[Music in Film|Music in film]]
* [[The Symphony and the Opera|Symphony and opera]]

{{ColBreak}}
=== Western music ===

* [[Brief History of Western Music]]
* [[Music of the Medieval Era]]
* [[Music of the Renaissance]]

{{ColBreak}}
* [[Music of the Baroque Era]]
* [[Music of the Classical Era]]
* [[Music of the Romantic Era]]
* [[Music of the 20th Century]]

{{EndMultiCol}}

==Music instruments==

{{MultiCol}}
=== [[String instruments]] ===

* [[Violin]]
* [[Viola]]
* [[Violoncello]]
* [[Double bass]]
* [[Fiddle]]
* [[Harp]]
* [[Guitar]]
** [[Classical guitar|Classic guitar (or ''&quot;Acoustic&quot;'' guitar)]]
** [[Electric Guitar|Electric guitar]]
** [[Bass guitar|Bass guitar]]
* [[Ukulele]]
* [[Banjo]]
* [[Mandolin]]
* [[Lute]]

{{ColBreak}}
=== [[Woodwind instruments]] ===

* [[Flute]]
* [[Oboe]]
* [[Clarinet]]
* [[Bassoon]]
* [[Saxophone]]
** [[Soprano Saxophone|Soprano saxophone]]
** [[Alto Saxophone|Alto saxophone]]
** [[Tenor Saxophone|Tenor saxophone]]
** [[Baritone Saxophone|Baritone saxophone]]
* [[Recorder]]
* [[Ocarina]]

===[[Brass instruments]]===

* [[Trumpet]]
* [[French horn]]
* [[Trombone]]
* [[Tuba]]
* [[Euphonium]]

{{ColBreak}}
=== [[Percussion instruments]] ===

* [[Concert Percussion|Concert percussion]] (Snare drum, crash cymbals, timpani, etc.)
* [[Drum set]]
* [[Mallet Instruments|Mallet instruments]] (Marimba, xylophone, vibraphone, chimes, etc.)
* [[Tabla]] (an Indian pair of drums)
* [[Pipe and tabor]]

=== [[Keyboard instruments]] ===

* [[Piano]]
* [[Organ]]

=== [[Topic: Voice | Voice]] ===

* [[Soprano]]
* [[Contralto]]
* [[Countertenor]]
* [[Tenor]]
* [[Baritone]]
* [[Bass (voice)|Bass]]

{{EndMultiCol}}

==Music resources==
[[wikibooks:Subject:Music|Wikibooks - Music]]{{MultiCol}}
=== Hands on ===

* [http://cigarboxguitars.com/workshops/How_To_Build_A_CBG.php Building a cigar box guitar]
* [[Blues basics]]
* [[Rock basics]]
* [[Wikiversity the Movie/music|Wikiversity the movie : music]]
* [[Jamming Online|Jamming online]]
* [[Experimental music]]
* [[Film scoring for Musicians|Practical lessons in film scoring]] (in conjunction with the [[Course:Practical narrative film editing|film editing course]])
* [[Digital Audio Workstation]]

=== Textbooks ===

* [[b:Music|Music theory]]
* [[b:Western Music History|Western music history]]
* [[b:Garageband Quick Tutorial|Garageband quick tutorial]]
* [[b:Triads|Triads]]
* [[b:Sound Recording|Sound recording]]

{{ColBreak}}
=== Open Source software ===

;For all operating systems

* [http://openmetronome.sourceforge.net/ Metronome]
* [http://sourceforge.net/projects/vtone/ Vtones] (cross platform midi editor)
* [http://audacity.sourceforge.net/ Audacity] (cross-platform sound editor ; a helpful tool for simple recordings and editing)
* [http://ardour.org/ Ardour] (digital audio workstation ; a great program for multi-track recording, mixing, mastering, etc.)
* [http://musescore.org/ Musescore] (cross-platform music notation software)
* [https://supercollider.github.io/ SuperCollider] (programming language and framework mainly for sound synthesis and algorithmic composition)

{{ColBreak}}
;For Linux

* [http://calf.sourceforge.net/ Calf Plugins] (very nice plugins : compressor, multichorus, reverb,etc.)
* [http://www.antcom.de/gtick/ GTick] (very nice and useful metronome for Gnome desktop)
* [[:w:LMMS|Linux MultiMedia Studio (''LMMS'')]]

=== External links ===

* [http://music.wikia.com/wiki/Music_Hub Music topics on Wikia]
* [[w:Wikipedia:Sound/list|Musical works available for download]]
* [[w:History of music|'History of music' on Wikipedia]]

=== Under Review ===
* [http://opensource.software.informer.com/download-opensource-guitar-tuner/ Guitar tuner] (Likely to be removed since this site has the chance to host malware.)
{{EndMultiCol}}

==Active participants==

''If you are an active participant in this school, you can list your name below. (this can help small schools grow and the participants communicate better)''
Please leave a timestamp - if it is more than a year old, there is potential for nomination to the inactive participants list.
*[[User:Kirby_-_Electrotechnics|Kirby]] (he/him), Banjo, May 2026

==Inactive participants==

*[[User:CQ|CQ]]
* Since 20 February 2012‎. Reviewed [[Portal:Pentatonic Impressionism (China Wu Sheng) in the view of Neo-classical Piano Techniques-training]] for Main Page News about 8 August 2019. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 19:58, 16 January 2020 (UTC)
* [[User:SelfieCity|SelfieCity]] 12 July 2021
*[[User:HappyCamper|HappyCamper]]
*[[User:Thierry613|Thierry613]]
*[[User:Bibeyjj|Bibeyjj]]
&lt;references /&gt;&lt;ref&gt;{{Citation|title=Audio-rate everything!|url=https://www.youtube.com/watch?v=RxHHJIDQC0A|date=2021-09-27|accessdate=2025-07-27|last=Omri Cohen}}&lt;/ref&gt;</text>
      <sha1>14u0337gfi0af4t37x1odnwoyhd3dtu</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Deletion policy</title>
    <ns>4</ns>
    <id>5572</id>
    <revision>
      <id>2810734</id>
      <parentid>2810529</parentid>
      <timestamp>2026-05-21T04:56:34Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Speedy */ Simplify wording</comment>
      <origin>2810734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19679" sha1="74a6h10a29a805pnlwhxdaxdi15kfn4" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be'''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>74a6h10a29a805pnlwhxdaxdi15kfn4</sha1>
    </revision>
    <revision>
      <id>2810735</id>
      <parentid>2810734</parentid>
      <timestamp>2026-05-21T04:56:53Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Speedy */</comment>
      <origin>2810735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19680" sha1="bfqmpc8i9kajup7kqntxf18ipn4pi3w" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources include:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>bfqmpc8i9kajup7kqntxf18ipn4pi3w</sha1>
    </revision>
    <revision>
      <id>2810739</id>
      <parentid>2810735</parentid>
      <timestamp>2026-05-21T05:07:31Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Criteria for speedy deletion */</comment>
      <origin>2810739</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19672" sha1="jcuysx6pu6e5lwzluiahkym3lp38ckn" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Types of deletion ==
=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>jcuysx6pu6e5lwzluiahkym3lp38ckn</sha1>
    </revision>
    <revision>
      <id>2810740</id>
      <parentid>2810739</parentid>
      <timestamp>2026-05-21T05:11:17Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Types of deletion */ Change to Deletion pathways and add an overview sentence</comment>
      <origin>2810740</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19868" sha1="pv69zz2crj3igybe0e3xgonevlmtyyg" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: speedy deletion (when noncontroversial) , proposed deletion (when a resource could be improved), and deletion request (when potentially controversial).

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>pv69zz2crj3igybe0e3xgonevlmtyyg</sha1>
    </revision>
    <revision>
      <id>2810741</id>
      <parentid>2810740</parentid>
      <timestamp>2026-05-21T05:15:59Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Deletion pathways */ + bullet-points + 1 sentence</comment>
      <origin>2810741</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20286" sha1="gi62d8s919iiyeax4rt4kqyp52pgufs" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''
*&lt;code&gt;{{tl|draft archive}}&lt;/code&gt; for proposal for '' moving under Draft:Archive/ '' (proposal is under construction)

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>gi62d8s919iiyeax4rt4kqyp52pgufs</sha1>
    </revision>
    <revision>
      <id>2810742</id>
      <parentid>2810741</parentid>
      <timestamp>2026-05-21T05:17:06Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Deletion templates */ Remove red link</comment>
      <origin>2810742</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20167" sha1="qsg9ufccixxeqt2egnc1hto8m229wzo" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace that cannot be directly used by teachers and instructors use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>qsg9ufccixxeqt2egnc1hto8m229wzo</sha1>
    </revision>
    <revision>
      <id>2810743</id>
      <parentid>2810742</parentid>
      <timestamp>2026-05-21T05:17:35Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Deletion templates */ Simplify 1st sentence</comment>
      <origin>2810743</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20110" sha1="8f1pnlck23gdvhq3lz1cof7f04ddhe6" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}
{{Shortcut|Deletion request|Dr|DR|Rfd|RfD}} 

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>8f1pnlck23gdvhq3lz1cof7f04ddhe6</sha1>
    </revision>
    <revision>
      <id>2810744</id>
      <parentid>2810743</parentid>
      <timestamp>2026-05-21T05:19:35Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Deletion request */ Remove shortcuts (now red links)</comment>
      <origin>2810744</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20065" sha1="e7a3msyzd85pzqjm7q3yhodrhte55ng" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

Note that the deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues.  

'''IMPORTANT:''' The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>e7a3msyzd85pzqjm7q3yhodrhte55ng</sha1>
    </revision>
    <revision>
      <id>2810746</id>
      <parentid>2810744</parentid>
      <timestamp>2026-05-21T05:21:32Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Proposed deletion */ Simplify/consolidate</comment>
      <origin>2810746</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20035" sha1="h02y7ttsr3hhxxtenanjqoxdgndyrvo" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].
*{{spaces|2}}''See also [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]''

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>h02y7ttsr3hhxxtenanjqoxdgndyrvo</sha1>
    </revision>
    <revision>
      <id>2810747</id>
      <parentid>2810746</parentid>
      <timestamp>2026-05-21T05:22:33Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Deletion request */ Move link to see also section</comment>
      <origin>2810747</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19913" sha1="qfm10900yrf19bdfn40dmyjxz9sfnby" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>qfm10900yrf19bdfn40dmyjxz9sfnby</sha1>
    </revision>
    <revision>
      <id>2810748</id>
      <parentid>2810747</parentid>
      <timestamp>2026-05-21T05:23:02Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* See also */ [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]</comment>
      <origin>2810748</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20011" sha1="tepfvqcjxfpuzjbecd6f8aatbp9pvla" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These templates are recommended for use by those involved with page deletions:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for large projects and difficult decisions
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts (templates and policy guidelines under construction)

'''Don't worry about selecting the &quot;correct&quot; template''', as more than one template may be appropriate. It is more important to provide assistance by flagging resources that you think shouldn't be on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>tepfvqcjxfpuzjbecd6f8aatbp9pvla</sha1>
    </revision>
    <revision>
      <id>2810750</id>
      <parentid>2810748</parentid>
      <timestamp>2026-05-21T05:28:01Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Copyedit nutshell</comment>
      <origin>2810750</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20002" sha1="iab171gdeledcvgv3s3vq91t4kuskkm" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These pathways and templates are recommended for nominating resources for deletion:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for potentially controversial decisions and projects with multiple pages
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts

'''Don't worry about selecting the &quot;correct&quot; pathway and template''', as more than one approach may be appropriate. It is more important to assist by flagging resources that you think shouldn't be retained on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>iab171gdeledcvgv3s3vq91t4kuskkm</sha1>
    </revision>
    <revision>
      <id>2810752</id>
      <parentid>2810750</parentid>
      <timestamp>2026-05-21T05:33:41Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2810752</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19996" sha1="ezfpm9mtg784s5c48q8v8kwgukrnacu" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These pathways and templates are recommended for nominating resources for deletion:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for potentially controversial decisions and projects with multiple pages
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts

Don't worry about selecting the &quot;correct&quot; pathway and template, as more than one approach may be appropriate. It is more important to assist by flagging resources that you think shouldn't be retained on Wikiversity.}}

This '''deletion guideline''' explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>ezfpm9mtg784s5c48q8v8kwgukrnacu</sha1>
    </revision>
    <revision>
      <id>2810753</id>
      <parentid>2810752</parentid>
      <timestamp>2026-05-21T05:34:18Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>guideline -&gt; policy</comment>
      <origin>2810753</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19978" sha1="1jy1ww40e13hee1wmuvbvizv5hjablf" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These pathways and templates are recommended for nominating resources for deletion:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for potentially controversial decisions and projects with multiple pages
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts

Don't worry about selecting the &quot;correct&quot; pathway and template, as more than one approach may be appropriate. It is more important to assist by flagging resources that you think shouldn't be retained on Wikiversity.}}

This policy explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources or specific revisions of a resource. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>1jy1ww40e13hee1wmuvbvizv5hjablf</sha1>
    </revision>
    <revision>
      <id>2810754</id>
      <parentid>2810753</parentid>
      <timestamp>2026-05-21T05:35:02Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Simplify sentence</comment>
      <origin>2810754</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19942" sha1="ijdvhvnshy3pbt61ug093zpvblij9ar" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These pathways and templates are recommended for nominating resources for deletion:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for potentially controversial decisions and projects with multiple pages
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts

Don't worry about selecting the &quot;correct&quot; pathway and template, as more than one approach may be appropriate. It is more important to assist by flagging resources that you think shouldn't be retained on Wikiversity.}}

This policy explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>ijdvhvnshy3pbt61ug093zpvblij9ar</sha1>
    </revision>
    <revision>
      <id>2810755</id>
      <parentid>2810754</parentid>
      <timestamp>2026-05-21T05:35:39Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2810755</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19964" sha1="owlwufbgsic9vyow6pgys81z57f8se1" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These pathways and templates are recommended for nominating resources for deletion:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for potentially controversial decisions and projects with multiple pages
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts

Don't worry about selecting the &quot;correct&quot; pathway and template, as more than one approach may be appropriate. It is more important to assist by flagging resources that you think shouldn't be retained on Wikiversity.}}

This policy explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion|Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''speedy deletion''' (when noncontroversial)
* '''proposed deletion''' (when a resource could be improved)
* '''deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
      <sha1>owlwufbgsic9vyow6pgys81z57f8se1</sha1>
    </revision>
    <revision>
      <id>2810756</id>
      <parentid>2810755</parentid>
      <timestamp>2026-05-21T05:36:06Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Deletion pathways */</comment>
      <origin>2810756</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19964" sha1="ozws43ku8cf38fog6buqq40b7d87yfj" xml:space="preserve">{{proposed|WV:D|WV:DEL|WV:DELETE}}
{{nutshell|While deletion is sometimes permissible when a good reason is given, resources can often blossom when participants are taught good wiki practices and how to implement concrete improvements. These pathways and templates are recommended for nominating resources for deletion:
* '''Speedy delete''' '''&lt;code&gt;{{tl|delete}}&lt;/code&gt;''' for pages you authored, as well as for pages that clearly don't belong in [[Help:Namespaces|mainspace]]
* '''Proposed deletions''' '''&lt;code&gt;{{tl|prod}}&lt;/code&gt;''' for pages that perhaps should be deleted (or moved to draft or user space) unless they are improved
* '''Deletion requests''' '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' for potentially controversial decisions and projects with multiple pages
* '''Move to draft or user space''' '''&lt;code&gt;{{tl|Pagemove announcement}}&lt;/code&gt;''' for research and student efforts

Don't worry about selecting the &quot;correct&quot; pathway and template, as more than one approach may be appropriate. It is more important to assist by flagging resources that you think shouldn't be retained on Wikiversity.}}

This policy explains when and why [[Wikiversity:Support staff|support staff]] may delete and restore resources. The [[Special:Log/delete|deletion log]] allows anyone to track deletions and restorations at Wikiversity. Deletion removes access while keeping resources and their development histories intact. Access can be restored through undeletion. Deletions and undeletions are discussed at [[Wikiversity:Requests for Deletion|Requests for Deletion]].

== Deletion pathways==
There are three main pathways for deletion: 
* '''Speedy deletion''' (when noncontroversial)
* '''Proposed deletion''' (when a resource could be improved)
* '''Deletion request''' (when potentially controversial)

These three options are described in more detail below. The practical steps for to nominate for deletion are described in the [[#Deletion templates|deletion templates]] section.

=== Speedy ===
Resources may be '''speedy deleted''' when a good reason is given by adding {{tlx|delete|''your good reason''}}. You may [[#Discussion|request community feedback]], or object with or without explanation by removing the speedy deletion suggestion from the resource. Resources may not be subject to speedy deletion when a reasonable objection is made or is likely because participants may be able to learn how to implement concrete improvements when taught appropriate practices.

====Criteria for speedy deletion====
{{shortcut|WV:CSD|WV:DB|WV:SD|WV:SPEEDY}}
A non-exhaustive list of possible reasons [[Wikiversity:Support staff|support staff]] may speedy delete resources:

;Common reasons
# '''Test page''' to practice use of the wiki software outside user space.  Experiment in the [[Wikiversity:Sandbox]]. Consider [[Wikiversity:Welcome templates|Welcoming the user]].
#'''[[Wikiversity:Vandalism|Vandalism]]''' and user pages for vandalism-only accounts.
#'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.
#'''Spam''' consisting of bulk automated creations, cross-wiki cross-posting, or repetitive external links with no educational value
#'''Repost''' with no concrete improvements to merit keeping. User should [[WV:RFD|request undeletion]] instead.
#'''Author request''' by an only contributor or only substantial contributor, or arising from an expressed consensus of contributors.
#'''Empty page''' which has been blanked or content not developed
#'''Empty category''' with no notice suggesting the category may be occasionally empty.
#'''Copyrighted work''' which does not satisfy [[Foundation:Terms of use|Wikimedia's Terms of Use]].
#'''Orphaned, broken, or unused redirect''' with no educational histories, no links, and either no appropriate targets or not likely to be searched for.
#'''No [[WV:WIW|educational objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.
#'''No [[Wikiversity:Research|research objectives]]''' or discussion in history. [[Wikiversity:Welcome templates|Welcome users]] and resources when likely to be expanded shortly.

;Less common reasons
# '''Abandoned resource'''
# '''Per [[WV:PROD|proposed deletion]]'''
# '''Per [[Wikiversity:Deletion policy#Discussion|request for deletion]]'''
# '''Basic maintenance''' such as repairing cut-and-paste moves, and moving resources over a redirect.
# '''Discussion''' about deleted resource where context is lost and becoming an independent resource is unlikely
# '''Resource [[m:transwiki|moved]]''' to another project due to [[Wikiversity:Scope|scope]], or file uploaded to [[commons:|Wikimedia Commons]] with history intact and links fixed.
#'''Unused file''' lacking either a license or a source may be deleted one week after uploader is notified.
#'''Ethical breach''' where a resource standing undeleted may cause harm
# '''Not English'''. See https://www.wikiversity.org to find where contributions may be appreciated
# '''History merge'''
# '''Deletion test'''

=== {{anchor|Proposed deletion}}Proposed deletion (prod)===
{{shortcut|WV:PROD|WV:PRD|WV:PD}}
Resources may be eligible for proposed deletion when educational objectives, learning outcomes, or research aims are scarce, and objections to deletion are unlikely. You may add {{tlxs|prod}} to the top of resources to propose deletion. Support staff may examine the resource after 90 days, and either delete the resource or remove the proposal. Anyone may object by removing the proposed deletion template from the resource, with an explanation on the resource's discussion page and/or in the edit summary. Anyone still considering that the resource should be deleted may [[#Discussion|discuss deletion]].

===Deletion request===
Use a deletion request if you think a developed page should be deleted or if the topic is complicated in some way. Then be prepared to engage in the discussion at [[Wikiversity:Deletion requests]].

== {{anchor|Deletion discussions|Discussion}}How to discuss ==
You should [[WV:AGF|ask questions and discuss concerns]] with participants, make an effort to [[Wikiversity:Be bold|resolve problems yourself]], and try [[#Alternatives to deletion|alternatives]] as appropriate before suggesting deletion for resources with substantial development histories and contributions from multiple participants. You may ask for guidance and advice, explain what you have done or will do, and discuss options at [[Wikiversity:Requests for deletion]]. Resources are generally kept when a community [[Wikiversity:Consensus|decision]] is inconclusive. Add {{tl|rfd}} to the top of resources to draw attention to the discussion. As a courtesy, you may also notify any participants directly.

Be concise and specific when suggesting solutions, because the preferred outcome is one participants can learn from to implement concrete improvements. On occasion, deletion might be permissible when productivity from a fresh start is likely and concerns relate to Wikiversity's [[Wikiversity:Mission|mission]], [[Wikiversity:Scope|scope]], [[Wikiversity:Process|process]], or [[Wikiversity:Policies|policy]] in some way. If you can raise concerns not addressed by a previous discussion, you may begin a new discussion at [[Wikiversity:Requests for Deletion|Requests for Deletion]]. Reasonable requests for resources to be restored during discussion are honored, where possible. Please only participate if you are willing to keep an open mind and make an effort to improve resources.

==Alternatives to deletion==
{{shortcut|WV:ATD}}
Deletion should be a last resort. Participants are encouraged to [[Wikiversity:Be bold|boldly]] consider constructive alternatives to deletion such as: 

{| class=&quot;wikitable sortable&quot;
! Issue
! Possible alternatives
|-
| valign=&quot;top&quot; |Outside [[WV:scope|Wikiversity's scope]]
| valign=&quot;top&quot; |Move to another Wikimedia project or more appropriate external platform. Some resources, such as personal essays, may be suitable for user space.
|-
| valign=&quot;top&quot; |Outside the scope of a learning project
| valign=&quot;top&quot; |Merge into another learning project or develop into a new learning project.
|-
| valign=&quot;top&quot; |Content dispute
| valign=&quot;top&quot; |Seek discussion, mediation, or wider community input before considering deletion.
|-
| valign=&quot;top&quot; |Foreign-language resource
| valign=&quot;top&quot; |Translate into English, move to the relevant language Wikiversity, or move to [[BetaWikiversity:|Beta Wikiversity]] if no Wikiversity exists in that language.
|-
| valign=&quot;top&quot; |Duplicate resource
| valign=&quot;top&quot; |Merge content, redirect to the primary resource, or move to a subpage with a clearer focus.
|-
| valign=&quot;top&quot; |Poorly organised resource
| valign=&quot;top&quot; |Reorganise sections, rename the resource, or restructure into subpages.
|-
| valign=&quot;top&quot; |Oversized resource
| valign=&quot;top&quot; |Split into smaller resources or reorganise into subpages.
|-
| valign=&quot;top&quot; |Undersized resource
| valign=&quot;top&quot; |Expand, merge into a broader resource, or redirect appropriately.
|-
| valign=&quot;top&quot; |Isolated content
| valign=&quot;top&quot; |Integrate into a broader learning project or create supporting pages.
|-
| valign=&quot;top&quot; |Low-quality content
| valign=&quot;top&quot; |Add cleanup, expansion, or context templates and encourage improvement.
|-
| valign=&quot;top&quot; |Insufficient or unclear educational value
| valign=&quot;top&quot; |Request clarification, improve the resource, or move to draft or user space for further development.
|-
| valign=&quot;top&quot; |Outdated resource
| valign=&quot;top&quot; |Update, archive, or mark as historical or superseded.
|-
| valign=&quot;top&quot; |Inactive learning project
| valign=&quot;top&quot; |Retain, archive, or reorganise if the project continues to have educational value.
|-
| valign=&quot;top&quot; |Abandoned draft
| valign=&quot;top&quot; |Move to draft or user space for future development.
|-
| valign=&quot;top&quot; |Unsuitable resource title
| valign=&quot;top&quot; |Rename or move to a more descriptive title.
|-
| valign=&quot;top&quot; |Unsourced or poorly sourced content
| valign=&quot;top&quot; |Tag for verification, request sources, or assist with adding references.
|-
| valign=&quot;top&quot; |Missing or incomplete licence information
| valign=&quot;top&quot; |Assist the user in providing the information or supply it if possible.
|-
| valign=&quot;top&quot; |Test or practice edits
| valign=&quot;top&quot; |Move to user space or sandbox space if there is constructive or educational value.
|-
| valign=&quot;top&quot; |Broken redirects
| valign=&quot;top&quot; |Redirect to a more appropriate target.
|-
| valign=&quot;top&quot; |Temporary concerns
| valign=&quot;top&quot; |Use maintenance, protection, discussion, or draftification templates until issues are resolved.
|}

==Deletion templates==
To remove resources from mainspace use the following templates:
*&lt;code&gt;{{tl|delete}}&lt;/code&gt; for ''speedy deletion''
*&lt;code&gt;{{tl|prod}}&lt;/code&gt; for a ''proposed deletion'' 
*&lt;code&gt;{{tl|dr}}&lt;/code&gt; for a ''deletion request''
*&lt;code&gt;{{tl|pagemove announcement}}&lt;/code&gt; informs editors that the page (and any subpages) are likely to be moved to either draft space or user space.  Wikiversity is a complex interaction between instructor teaching materials, student efforts, and even research. Mainspace should focus on teaching, learning, and research resources.
*&lt;code&gt;{{tl|draftify}}&lt;/code&gt; for ''proposed moving to the Draft: namespace''

===Speedy deletion===
*Use &lt;code&gt;{{tl|delete}}&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{delete|(with optional explanation)}}&lt;/nowiki&gt;&lt;/code&gt;
Use '''Speedy deletion''' for uncontroversial deletions.  Placing '''&lt;code&gt;&lt;nowiki&gt;{{delete|&lt;suggested reasons go here ...&gt;}}&lt;/nowiki&gt;&lt;/code&gt;''' at the top of a page creates this banner:
&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{ambox
|type=speedy
|text=This {{lc:{{NAMESPACE}}}} page may qualify for [[Wikiversity:Deletion policy#Speedy|speedy deletion]] because: &lt;suggested reasons go here...&gt;&lt;br/&gt;{{#ifeq:{{{self}}}
|yes
|If you disagree, please remove this notice.&lt;br /&gt;
|If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}}
 |{{TALKSPACE}}
 |this discussion page
 |[[{{TALKPAGENAME}}|the discussion page]]
}}, after adding &lt;span style=&quot;font-family:'Lucida Console', monospace&quot;&gt;{{Tl|hangon}}&lt;/span&gt; to the top of the {{lc:{{NAMESPACE}}}}. This will alert [[Wikiversity:Support staff|support staff]] to your intention, and may permit you the time to write your explanation.  Try to make your explanation more eloquent and less hysterical than the example shown above.
}}

&lt;span class=&quot;plainlinks&quot;&gt;''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletion policy]]. ''&lt;/span&gt;
}}
&lt;!--Must delete this: [[Category:Candidates for speedy deletion|{{PAGENAME}}]]--&gt;
&lt;!-- Ends the delete demonstration--&gt;

The most common use of ''speedy deletion'' is for pages written by a single author and which include copyright violations, pseudoscientific claims, inappropriate language, advertisements, or slander or other efforts to harass living persons. 

Anything that potentially causes harm should be immediately deleted, and ''speedy deletion'' is the fastest way to nominate a page for removal. But a ''speedy deletion'' is also the most time-consuming to revert (if the resource actually gets deleted). A good rule of thumb is never to speedy delete a resource that might be [[wikt:salvable|salvable]] or has had active edits in the past three years. Also, hesitate before you ''speedy delete'' a project or a large number of subpages, as reverting such deletions can be extremely time-consuming.

===Proposed deletion===
*Uses &lt;code&gt;{{tlxs|prod}}&lt;/code&gt;

'''Proposed deletion''' (or &quot;prod&quot;) is for pages that may or may not belong in mainspace. It allows editors to debate possible deletion on the talk page and ponder other options such as moving to the user or draft namespace. ''Prod'' draws attention and can result in an effort to improve the resource. ''Prod'' places the discussion about whether to delete on the talk page. ''Prod's'' greatest disadvantage is it causes projects to remain in mainspace until the question is resolved. A simple remedy for this is to move the project into user or draft space until an outcome has been decided. Support staff can move a resource with up to 100 subpages.&lt;ref&gt;Moving a resource with 300 subpages takes about three times as long as moving one with 100 subpages, with the added feature that the moved resources are now required to have 3 top pages (with 100 subpages each)&lt;/ref&gt; To understand how ''prod'' works, suppose that on January 1, 2000 you placed '''{{tlxs|prod}}''' at the top of a page to create this banner:

&lt;!-- THIS TEXT WAS CREATED BY CALLING THE TEMPLATE WITH tlxs--&gt;
{{Ombox
| type  = delete
| image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]]
| text  = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|January 1, 2000 +90 days}}''' may allow it to be [[WV:PROD|kept]]. &lt;br&gt;
You may remove &lt;nowiki&gt;{{proposed deletion}}&lt;/nowiki&gt; from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]].
}}

&lt;div class=&quot;center&quot; style=&quot;width:auto; margin-left:auto; margin-right:auto;{{#if:  | {{{style}}};}}&quot;&gt;[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]&lt;/div&gt;
&lt;!-- Ends the tlxs|prod demonstration--&gt;

The deadline for improving the page is 90 days after the template was inserted. This gives editors time to improve the page or defend its virtues. The proper venue for all discussions is the [[Help:Talk page|talk page]] of the article under consideration.

===Deletion request===
* Uses &lt;code&gt;{{tl|dr}}&lt;/code&gt;

Use '''Deletion request''' if you think a developed page should be deleted or if the topic is complicated in some way. Begin by placing '''&lt;code&gt;{{tl|dr}}&lt;/code&gt;''' at the top of the page to create:
{{ambox
|image = [[File:User-trash.svg|50x50px|link=]]
|text='''Please [[Wikiversity:Requests for Deletion#{{FULLPAGENAME}}|share your thoughts]] about whether to [[Wikiversity:Deletion policy|keep this resource or not]]'''. Resources are likely to remain at Wikiversity when you [[Wikiversity:Be bold|boldly]] address reasonable concerns through concrete improvements. We encourage you to give resources a chance to receive fair reviews and concrete improvements by keeping nomination notices intact.}}
{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}|''Link to any subpages this page might have'']]}}

Click the words [[Wikiversity:Requests for Deletion|'''share your thoughts''']] in the box. This will lead to a page called [[Wikiversity:Requests for Deletion]]. Follow the instructions for starting a discussion. Explain why you think the page should be deleted. Be prepared to engage in discussion about the page's value. At some point the discussion will be closed. Sometimes the decision may be to place the article in [[Wikiversity:Drafts|draft space]] or on a subpage of their [[Help:User page|user page]].

===Pagemove announcement===
*Uses &lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;
The advantage of moving material into the user or draft namespace is that less time is spent discussing and more time cleaning up. Wikiversity has a limited staff of volunteers, and the quicker decisions are made, the more that can be done. Too many obscure pages in mainspace gives others a bad impression of Wikiversity. On the other hand, people should be encouraged to write on Wikiversity about all sorts of topics. The goal is to separate materials recommend for teachers, learners, and researchers from the discovery and exploration done by Wikiversity users.
&lt;!-- No link or evidence for this: A procedure and guidelines for moving mainspace resources into user and draft space is under construction. --&gt;

'''&lt;code&gt;&lt;nowiki&gt;{{subst:Pagemove announcement}}&lt;/nowiki&gt;&lt;/code&gt;''' is used to communicate with all the page's main authors' talk pages and looks like this:
{{Robelbox|theme=8|title=Removal of a project from mainspace|width=100%}}
To ''Somebody's username'':

The project '''&lt;nowiki&gt;[[Some page]]&lt;/nowiki&gt;''' has been moved to '''&lt;nowiki&gt;[[User:Somebody's username/Some page]]&lt;/nowiki&gt;'''  

A tentative decision has been made to remove this project from [[Wikiversity:namespaces|mainspace]]. If this move is not under discussion at '''[[Wikiversity:Requests for Deletion]]''', you may make a request regarding this project at '''[[Wikiversity:Request custodian action]]'''. 

Yours truly, --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 06:07, 20 December 2022 (UTC)
{{Robelbox/close}}

== See also ==
* [[MediaWiki:Deletereason-dropdown]] - List of common deletion reasons displayed in the deletion interface
* [[:Category:Candidates for speedy deletion]] - List of current resources suggested for [[#Speedy|speedy deletion]]
* [[:Category:Proposed deletions]] - List of current resources [[#Proposed|proposed]] for deletion
* [[:Category:Requests for Deletion]] - List of resources with active deletion [[#Discussion|discussions]]
* [[w:Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_democracy|Wikipedia is not a democracy]]

{{Official policies}}
{{Proposed policies}}
&lt;!-- {{subpagesif}} There are currently no subpages --&gt;

[[Category:Wikiversity deletion]]</text>
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  <page>
    <title>Religious Law</title>
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        <username>Atcovi</username>
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      <minor />
      <comment>project box(es)</comment>
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      <text bytes="226" sha1="jlbra0rm2v0s1lipppti39y7v5lznvm" xml:space="preserve">{{law}}
{{religion}}
== Resources ==
* [[The Laws of the Pharisees]]
* [[Shariah Law]]
* [[Agape]]
* [[Church Doctrine]]
* [[The Ten Commandments]]

{{subpagesif}}
[[Category:Law learning projects]]
[[Category:Law by subject]]</text>
      <sha1>jlbra0rm2v0s1lipppti39y7v5lznvm</sha1>
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    <title>Planning economic development</title>
    <ns>0</ns>
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      <comment>/* Non-profit */</comment>
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      <text bytes="7706" sha1="t0uj9vb71jxjsybpkayqbnsmv7s8ig0" xml:space="preserve">This course will examine factors in planning economic development. Through this course you will hopefully leave with a better sense of how to plan for quality economic development. 

There are a multitude of different perspectives on economic development that fall into the realms of economic, political and philosophical.

Questions that will be explored are:
* How can economic development best be stimulated? 
* What sort of business regulations are most conducive to quality economic development?
* How can regulations be developed to maximize the benefit for both the people of the state and the businesses and their customers?

==What is economic development?==
'''Definition'''
Economic development has several definitions from local to global perspectives. Professor of Economics and Public Policy Alan Deardorff at the University of Michigan as part of his International Economics Glossary calls it: &quot;Sustained increase in the economic standard of living of a country's population, normally accomplished by increasing its stocks of physical and human capital and improving its technology.&quot;&lt;ref&gt;[http://www-personal.umich.edu/~alandear/glossary/e.html International Economics Glossary ]&lt;/ref&gt; At the local level, the term is brought to a more reachable level. For example the UrbanPlan curriculum states: &quot;Economic development—A term generally applied to the expansion of a community’s property and sales tax base or the expansion of the number of jobs through office, retail, and industrial development.&quot;&lt;ref&gt;[http://www.urbanplan.org]&lt;/ref&gt; An interesting Cornell article expands: &quot;Economic development is typically measured in terms of jobs and income, but it also includes improvements in human development, education, health, choice, and environmental sustainability. Business and economic developers in the US are increasingly recognizing the importance of quality of life, which includes, environmental, and recreational amenities, as well as social infrastructure such as child care, in attracting and retaining businesses in a community.&quot;&lt;ref&gt;[http://government.cce.cornell.edu/doc/html/MethodologyGuide_TermsUsed.htm Cornell]&lt;/ref&gt; In each of these definitions, the focus is on growth of the physical and social sphere of life. As well there is an inherent goal, that such growth achieves a greater &quot;standard of living&quot; or &quot;quality of life.&quot; Growth and improvement, of course, not simply endless building such as that of residential suburbia. However the field does presume one common fact, that what ''is'' there is not providing what that community needs. 

'''In government structure'''
Economic development (ED) is seen both as a policy and a profession. In the United States, most local governments have an economic development authority that oversees and guides enterprise in states and cities. Many states have multiple layers of such groups. For example, a neighborhood might have it's own non-profit to help small businesses establish storefronts. The municipality's economic development department would help major corporations locate into city limits. Most cities have metropolitan regions, and such regional authorities can promote entire areas of a state for new companies. The state's workforce and employment department would then overlap all of these, tracking job growth and ensuring Federal funds may be available for job programs and assistance. ED many times is an expected part of government function because cities and regions are constantly competing for jobs which no longer need a specific location. 

'''As government policy'''
As policy, ED is frequently noted in the news as a function of a country's government to improve the welfare of its citizens by providing and literally building opportunities. The skyscrapers and dams being fervorently built in China's coastal cities and rural west have become a symbol of ED. The term started for American cities in the advent of suburbia in the 1970s but had not exactly entered American politics until the economic boon of the 1990s. ED has traditionally been applied to major projects such as a new industrial zone or enclosed shopping mall as well as waterworks projects and freeway expansion. However suburbia by the 1990s began to realize that the capitalist micro-economies of downtowns were not going to remake themselves and the term gained full footing to ensure stability in new cities by carefully planning and plotting the location of potential retail, services, and office. ED has also become familiar with medical and hospitality industries, seeing hospital campuses and hotels as valuable as an office tower. The main goal with ED as government policy is that jobs must grow in the end, much like how private companies ultimately wish to gain profits from new investment. 

'''As a profession'''
As a profession, ED Directors and Business Specialists work with business owners and much like courting deals in the private sector, will try to provide opportunities to entrepreneurs. These may include qualifying special new business loans, offering tax breaks on a piece of land, or ensuring planning officials can compromise to approve a project. While ED personnel are generally &quot;on the ground,&quot; they also do extensive research and quantitative analysis as to potential sites which may accommodate future employers. Regularly they perform many urban planning and community development functions such as identifying properly zoned areas for commercial or industrial and the accompanying codes and variances that could suit a business model. With these goals, ED staff may also influence planning decisions and encourage the establishment of Enterprise Tax Zones which specifically encourage businesses to locate in a particular geographic area. 

===The city and the hinterland===
The basis for urban economies is in understanding the relationship of cities and the hinterland.  The hinterland has been used since the 20th century by geographers to describe rural land, the &quot;empty&quot; and &quot;wild&quot; space between cities. 
Eugene van Cleef would better define it in &quot;Hinterland and Umland,&quot; tracing its Germanic routes as the land extending from the coast.&lt;ref&gt;{{cite book|title=Hinterland and Umland|author=Eugene van Cleef|volume=Geographical Review, Vol. 31, No. 2 (Apr., 1941), pp. 308-311   (article consists of 4 pages)|publisher=Published by: American Geographical Society|url=http://www.jstor.org/pss/210211}}&lt;/ref&gt; The best equivalent is &quot;back country.&quot; Though with a slight negative connotation, the hinterland has become an appropriate term to describe land areas that do not necessarily have a city or urbanized function but are not necessarily rural lands. For example much of the central to western United States is unincorporated or not in use at all due to natural features.  As well hinterland encompasses all natural lands even those in protected status.   

The need to understand the relationship between the city (meaning both urbanized and metropolitan areas) and the hinterland, is helpful to understanding the cycle of urban economies and flow of investment and assets. ((more text to be added))

===Urban economies: cycle flow and assets===
===Metropolitan spheres of influence===

==Planning for enterprise==
===Types of businesses===
===Infrastructure===
===Urban conditions8===
===Regulations===
====Zoning code====

==Techniques== 

===Private sector===
====Retail====
====Commercial====
====Industrial====

===Non-profit===
====Medical facilities====
====Arts and cultural institutions====
====Advocacy groups====

===Community groups===
====Neighborhood investment====


[[Category:Urban Studies and Planning]]
[[Category:Economic development]]
[[Category:Sustainability]]</text>
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    <title>Aviation Weather</title>
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      <text bytes="9807" sha1="n2njicc6q7x4b9q08haip6jkegpa3rz" xml:space="preserve">{{environmental science}}
== Online Courses ==

[http://www.flightcentral.net/gs-lessons/wxtheory/player.html Aviation Weather Theory]

==The pilot's weather tools==
To accurately gather [[weather data]] to ensure a successful [[flight]], pilots have access to many different resources that inform them of what the [[weather]] is up to.

== METARs ==
Every pilot and even student for that matter has been exposed to a [http://www.pilotpedia.com/wiki/index.php/METAR METAR]. Roughly translated from the French as METeorlogical Aviation Routine weather Reports, a METAR is the hourly surface weather observation issued 5 minutes before the hour. It is available to the aviation community and used by the National Weather Service to determine an airport's flying conditions ([[Instrument Flight Rules|IFR]], [[Marginal Visual Flight Rules|MVFR]], [[Visual Flight Rules|VFR]]). You can obtain these from a variety of places both online and by phone. 

This is the basic knowledge of what ground school consists of on this topic other than deciphering its acronyms. Although as a pilot, sometimes its nice to know a little about how these services came to be without going too much into detail. Funny enough, the change to our current acronym of METAR is fairly recent. 

====History====
Before the current interpretation of weather data, there were two formats, and, in rough terms, it was &quot;us and them&quot; or rather, to be more precise, the North American countries reported weather differently from the rest of the world. North Americans were using SAO or &quot;Surface Aviation Observation&quot; (which was adopted in the 1950s), while the rest of the world was using the currently known METAR or &quot;METeorological Aviation Routine weather reports&quot;. The FAA, which determines aviation requirements in the US, became increasingly aware that expanding numbers of international flights and pilots alike was creating a strong need to standardize weather report interpretations internationally. 

The [[National Weather Service]] standardized the weather reports into what we now know as METARs. To reduce the stress on US aviation citizens, the metric system was kept to a minimum: for example, the winds were kept in knots instead of meters per second, visibility in miles, altimeter settings in inches of mercury instead of hectopascals (hPa), and RVR continued in feet. Temperatures, however, are converted to Celsius to allow for better conversions. The biggest change is simply the order in which elements are reported.

====Translating METARs====
So in a sense, although the attempt at standardizing weather code from SAO to METAR is apparent, the actual translation is not substantial. Oh and for those of you who are students or pilots that don’t like translating code, there is a reason and a cure. First off do realize that METARs are always originated in code and probably will be for a while. The reason seems to be that with the vast amount of changes and updates to weather reports would overload the system. The great news is that now it is possible to translate the code online either through request from sites such as NOAA or by looking up the code yourself which will put it in simple English. Sometimes this helps those who are new to learn the code and not miss anything during preflight weather preparations. 

====A closer look at the METAR====
The following example is a METAR taken from Vancouver International Airport ([[ICAO airport code|CYVR]]) in Vancouver, British Columbia:

METAR CYVR 120200Z 14021G27KT 20SM -SHRA FEW030 BKN058 OVC090 10/06 
A2982 RMK SC2SC5AC2 PCPN VRY LGT SLP098= 

Like mentioned above, METARs are observed and posted on an hourly basis. An airport can also issue an METAR that is not on the hour, this observation is called a SPECI. SPECI is abbreviated for special because the report was issued based on a significant change in conditions such as a violent temperature change, cloud layer movement, precipitation, visibility, etc. Looking back at the Vancouver METAR, the observation can be translated as such.


'''METAR''': Simply indicated that the observation is a METAR observation.

'''CYVR''': This is the aerodrome ident of which the observation originates, in this case, the aerodrome is Vancouver International.

'''120200Z''': This is the time of which the METAR was issued, the first two numbers are the date, so ''12'' would mean the twelfth day of the month. The ''0200Z'' is the time indicated in UTC, in aviation, it is indicated as Zulu time hence the Z on the end. The time 0200 can be translated to 2:00.

'''14021G27''': This is the current wind observation. The first three numbers indicate the wind direction '''''in degrees true''''' while the following numbers indicate the wind speed measured in knots. From this figure we gather that the wind is at 140 degrees true and is at 21 knots gusting to 27 knots.

'''20SM''': Indicates the visibility at the airport; in this case, the visibility is 20 Statute Miles.

'''-SHRA''': Means that there is rain showers at the airport, the negative sign at the beginning indicates the severity of the precipitation, a negative sign meaning light, no sign at all meaning moderate, a positive(+) sign meaning heavy.

'''FEW030 BKN058 OVC090''': These are the current cloud layers observed. ''FEW030'' is few clouds at 3000 feet ASL, ''BKN058'' is broken clouds at 5800 feet ASL, and ''OVC090'' is overcast clouds at 9000 feet ASL. ''ASL'' is abbreviated for above sea level, and you add two zeros on the end of the numbers to receive the actual altitude of the clouds.

'''10/06''': Current temperature and dewpoint, Temperature is 10 degrees celsius while the dewpoint is 6 degrees celsius.

'''A2982''': Altimeter setting observed, in this case 29.82 Hg/m 

'''RMK SC2SC5AC2 PCPN VRY LGT''': Any remarks which the observation has. ''SC2SC5AC2'' indicates that in respect to the observed cloud layers, the clouds are composed of 2 ocatas of strato cumulus clouds, 5 octas of stratocumulus clouds, and 2 octas of alto-cumulus clouds. ''PCPN VRY LGT'' indicates that precipitation is very light.

'''SLP098''': Indicates sea level pressure.

==== Quiz ====
&lt;quiz display=&quot;simple&quot;&gt;

{METARs are roughly translated from the French as _________. 
|type=&quot;[]&quot;}

+ METeorological Aviation Routine weather Reports 
|| Correct! This is the accurate translation of METAR from French.
- METeorological Airline Regular weather Reports 
|| Incorrect. Remember, METAR stands for METeorological Aviation Routine weather Reports.
- METeorological Airport Regular weather Reports 
|| Incorrect. The correct translation is METeorological Aviation Routine weather Reports.

{Before the current interpretation of weather data, North Americans were using _________ while the rest of the world was using METAR. 
|type=&quot;[]&quot;}

+ SAO or Surface Aviation Observation 
|| Correct! SAO stands for Surface Aviation Observation.
- Surface Aircraft Observation 
|| Incorrect. The correct term is Surface Aviation Observation.
- Standard Aviation Observation 
|| Incorrect. Remember, it's Surface Aviation Observation.

{The FAA became increasingly aware that expanding numbers of international flights and pilots was creating a need to _________. 
|type=&quot;[]&quot;}

+ standardize weather report interpretations internationally 
|| Correct! The FAA aimed to standardize weather report interpretations on an international scale.
- localize weather report interpretations 
|| Incorrect. The FAA's goal was to standardize, not localize, weather report interpretations.
- diversify weather report interpretations 
|| Incorrect. The aim was to standardize, not diversify, the interpretations.

{METARs are always originated in code because _________. 
|type=&quot;[]&quot;}

+ the vast amount of changes and updates to weather reports would overload the system 
|| Correct! Using code for METARs prevents the system from being overwhelmed by frequent changes and updates.
- it's easier for pilots to read 
|| Incorrect. The primary reason is to prevent system overload, not ease of reading.
- it's an international standard 
|| Incorrect. While METARs are standardized, the main reason for using code is to prevent system overloading.

{In the METAR example from Vancouver International Airport, &quot;14021G27&quot; indicates _________. 
|type=&quot;[]&quot;}

+ the wind is at 140 degrees true and is at 21 knots gusting to 27 knots 
|| Correct! The wind direction is 140 degrees true, with a speed of 21 knots and gusts up to 27 knots.
- the wind is at 140 knots and is gusting from 21 to 27 degrees 
|| Incorrect. The first three numbers indicate wind direction, and the following numbers indicate wind speed and gusts.
- the wind is at 21 degrees true and is at 140 knots gusting to 27 knots 
|| Incorrect. The wind direction is given first, followed by speed and gusts.

{In the METAR example, &quot;FEW030 BKN058 OVC090&quot; refers to _________. 
|type=&quot;[]&quot;}

+ the current cloud layers observed 
|| Correct! FEW030 means few clouds at 3000 feet, BKN058 means broken clouds at 5800 feet, and OVC090 means overcast clouds at 9000 feet.
- the visibility range in miles 
|| Incorrect. These codes specifically refer to cloud layers, not visibility.
- the temperature and dewpoint 
|| Incorrect. These codes describe cloud layers, not temperature or dewpoint.

&lt;/quiz&gt;

== Resources ==

{{Wikisource|Aviation Weather AC 00-6A}}
*http://www.nws.noaa.gov/oso/oso1/oso12/overview.htm - National Weather Services 
*http://www.alaska.faa.gov/fai/afss/metar%20taf/metintro.htm (Also a good source for additional info and abbreviation translations.)
*http://www.nwstc.noaa.gov/METEOR/AvnOps/aoc_webpage.htm NOAA National Weather Service Training Center.
*http://adds.aviationweather.noaa.gov Aviaition Digital Data Service Aviation Weather Center

[[Category:Aviation]]
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* {{user|CQ}}
* {{user|IanVG}}
* ...''you''...

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    <title>Naval architecture</title>
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      <comment>/* Content summary */ subpages</comment>
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      <text bytes="2911" sha1="10mlmg2scm3cszjg15jldg92u5aas5g" xml:space="preserve">'''[[Portal:Marine engineering|Wikiversity Department of Marine Engineering]]'''

{{engineering}}
{{tertiary}}
{{course}}
==Content summary==
We discuss primarily the ship as a system, and as a link in the transportation system. Introduce the participant to the complexities in the marine environment and the mechanisms involved in the ship system. The participant is also introduced to the various terms used in naval architecture.
[[Image:Neue_Planet_von_vorn.jpg|thumb|right|300px|Research ship ''Planet'' of the German Navy, a [[W:SWATH|SWATH]] design which evolved from the catamaran concept.]] 
* [[/Mechanics of Floating Bodies/]]
* [[/Ship Resistance/]]
* [[/Ship Propulsion/]]
* [[/Ship strength/]]
* [[/Vibration in Structures/]]
* [[/Seakeeping/]]
* [[/Manouvreing/]]
* [[/Marine Engineering/]]
* [[/Ship Design/]]
* [[/Design of High Speed Craft/]]
* [[/Ship Construction/]]
* [[/Production Planning/]]

==Goals==
To introduce the terms commonly used in naval architecture and introduce the ship as a system.


==[[Portal:Learning Materials|Learning materials]]==
Learning materials and [[Portal:Learning Projects|learning projects]] are located in the main Wikiversity namespace. Simply make a [[link]] to the name of the lesson (lessons are independent pages in the [[Wikiversity:Namespaces|main namespace]]) and start writing! 

You should also read about the [[Wikiversity:Learning]] model.  Lessons should center on learning activities for Wikiversity participants. Learning materials and learning projects can be used by multiple projects. Cooperate with other departments that use the same learning resource.
* ...

===Texts===
* Principles of Naval Architecture (2nd Rev.); Lewis, E.V , 1989, SNAME New York
* Basic Ship Theory (5th Edition); K.J.Rawson and E.C.Tupper, 2001, Butterworth-Heinemann

===Naval Architecture Lessons===
* '''Hull Design'''
# Curves of form
# Coefficients of form
* '''Stability'''
# Buoyancy
# Righting arm moments
* '''Hydrodynamics'''
# Resistance
# Propulsion
* '''Loading'''
* '''Behavior '''
# Slamming
# Hogging
* '''Box Girder Theory'''
* '''Vessel Design'''
# Models &amp; Scaling

===Assignments===
====Readings====
Each activity has a suggested associated background reading selection.
* '''Wikipedia article:''' [[w:Naval architecture|Naval architecture]]
* '''Wikibooks:''' [[Wikibooks:Subject:Marine engineering|Marine engineering]]

==References==
Additional helpful readings include:
* Elements of Ocean Engineering; Randall, Robert E., 1997, SNAME New York
* Principles of Naval Architecture (2nd Rev.); Lewis, E.V., 1989, SNAME New York
* Basic Ship Theory (5th Edition); K.J.Rawson and E.C.Tupper, 2001, Butterworth-Heinemann

==Active participants==
Please join in and post your designs!! Active participants in this [[Portal:Learning Projects#Learning Groups|Learning Group]]
* ...

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      <text bytes="1235" sha1="3x9rnmj2zrpro4q6yt2z84gdpj4v437" xml:space="preserve">{{Prod|page has not been developed since 2007 and doesn't seem to be linked to any substantial content}}
This page is about '''modeling Wikiversity through [[Activity Theory]]'''

Example model 1

SUBJECT: Wikiversity participants

OBJECT: Learning materials

MEDIATING ARTIFACTS: Wiki

RULES: Policies &amp; guidelines in Wikiversity (and, to an extent, Wikimedia)

COMMUNITY: Wikimedia (projects + ‘organisation’), bloggers, OER projects, etc.

DIVISION OF LABOUR: Open and distributed (?), organised by “schools”, subject disciplines (?)

OUTCOME: Learning experiences


Questions
• Where are the barriers to technology in this model?
• Where is identity?
• Where are theories of learning, and learning material production?


Example model 2

SUBJECT: Wikiversity participants

OBJECT: Wiki

MEDIATING ARTIFACTS: Theories/principles of educational and informational web design

RULES: ?

COMMUNITY: Designers, Wikimedia (as above), bloggers, OER projects…

DIVISION OF LABOUR: Open and distributed (?), organised by “taskforces”

OUTCOME: Increased participation


Questions
What are the rules in this model? Is it theories of participation (and hence identity, access, etc.)?

Cormac Lawler; Friday, 25th May, 2007</text>
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      <text bytes="5875" sha1="tgf1x2dz3tld1u4lks6sd43vnykh547" xml:space="preserve">{{medicine}}
{{law}}
Welcome to the Wikiversity learning project for '''medical practice and the law'''. This project allows Wikiversity participants to explore what constitutes &quot;medical practice&quot; and what is legally recognized as &quot;practicing medicine without a license&quot;. The emphasis here is in making clear how Wikiversity participants can discuss medical topics without being accused of &quot;practicing medicine without a license&quot;.

==Activity: improve the Wikiversity [[Wikiversity:Medical disclaimer|medical disclaimer]]==
The basic issue is that Wikiversity does not provide medical advice and nothing &quot;said&quot; at Wikiversity should be taken as constituting medical advice. 

===Draft changes here===
The following is a work space for drafting new modifications of [[Wikiversity:Medical disclaimer]].
&lt;div style=&quot;text-align: center; font-size: x-large; padding: 1em;&quot;&gt;WIKIVERSITY DOES NOT GIVE MEDICAL ADVICE&lt;/div&gt;

Wikiversity encourages wiki participants to explore and discuss [[w:list of medical topics|medical topics]]; however, '''no warranty whatsoever''' is made that any of the Wikiversity content related to medical topics is accurate. There is absolutely no assurance that any statement contained or cited in a Wikiversity webpage touching on medical matters is true, correct, precise, or up-to-date.  The overwhelming majority of Wikiversity content is written, in part or in whole, by nonprofessionals.  Even if a statement made about medicine is accurate, it may not apply to you or your symptoms.

The medical information provided on Wikiversity is, at best, of a general nature and cannot substitute for the advice of a '''medical professional''' (for instance, a qualified doctor/physician, nurse, pharmacist/chemist, and so on).  '''Wikiversity does not provide medical advice.''' Wikiversity attempts to discourage participants from providing medical advice. Wikiversity participants should remove medical advice from Wikiversity webpages when they see it or tell the [[Wikiversity:Request custodian action|custodians]]. However, most discussions of medical topics do not constitute medical advice. 

None of the individual contributors, system operators, developers, sponsors of Wikiversity nor anyone else connected to Wikiversity can take any responsibility for the results or consequences of any attempt to use or interpret as medical advice any of the information presented on this web site. If you find Wikiversity content that you believe constitutes medical advice, please discuss your concerns with other Wikiversity participants.

'''Nothing on Wikiversity.org or included as part of any project of [[w:Wikimedia Foundation|Wikimedia Foundation]] Inc., should be construed as an attempt to offer or render a medical opinion or otherwise engage in the [[w:medicine|practice of medicine]].'''

==Legal definitions of medical practice and medical advice==
There are some characteristic elements in the practice of medicine. The following examples suggest how to discuss medical topics without giving medical advice.
*&quot;[[w:Medical prescription|Prescribing]], dispensing, or administering any medicinal drug&quot; (example, [http://www.leg.state.fl.us Florida state law]). In wiki discussions (such as those at [[Wikiversity:Help desk|help desk]]), do not say, &quot;You should take drug X for condition Y,&quot; but you might say, &quot;According to the [http://www.bt.cdc.gov/agent/anthrax/treatment/ CDC], [[w:Ciprofloxacin|Ciprofloxacin]] is one drug that is used to treat [[w:Anthrax|Anthrax]]. If you are describing a real world health problem, consult a doctor.&quot; Drugs are just one form of medical treatment; there others such as [[w:Surgery|surgery]]. At Wikiversity we do not tell people to use a particular treatment, but we do have [[learning resource]]s that discuss which treatments are used for particular medical problems. 
*[[w:Diagnosis|Diagnosis]] of a physical, mental or emotional condition. In wiki discussions, do not say, &quot;Tell us more about your symptoms so that we can figure out your health problem&quot;. However, you might say, &quot;That list of symptoms is similar to the list found at [[Diabetic Foot Exam]]. If you are describing a real world health problem, consult a doctor. You cannot get a useful medical diagnosis on the internet.&quot;
*'''Patient-physician relationship'''. Establishing a [http://www.ama-assn.org/ama/pub/category/4607.html patient-physician relationship] is a normal part of practicing medicine. Wikiversity participants might fruitfully have informal student-teacher relationships when they exchange information about a medical topic, but Wikiversity participants should not establish any kind of relationship that begins to resemble a patient-physician relationship. Direct people who ask for medical advice to the [[Medical advice tutorial]].
*[[w:Prognosis|Prognosis]]. A common part of the practice of medicine involves physicians providing the best available information about the expected course of an illness. If a [[Wikiversity:Help desk|help desk]] participant asks a question such as, &quot;I have lung cancer, how long will I live?&quot; then do not try to answer the question....just tell the person to ask a doctor. There is no point in accepting such questions framed as a patient seeking a prognosis. However, questions such as, &quot;What is the average life expectancy following a diagnosis of lung cancer,&quot; can be answered by [[Wikiversity:Cite sources|citing]] [[Wikiversity:Reliable sources|reliable sources]] such as [http://www.cdc.gov/cancer/lung/ the CDC], [http://www.cancer.gov/cancertopics/wyntk/lung NCI] and the [http://www.nlm.nih.gov/medlineplus/ency/article/000122.htm NIH].

==Related resources==
*[[Medical advice tutorial]] - for wiki participants who ask for medical advice

[[Category:Evidence-based medicine]] 
[[Category:Legal ethics]]
[[Category:Wikiversity culture]]
[[Category:Medical School Program]]</text>
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      <text bytes="1759" sha1="nlvnpgzufvq8x172gjamyfe49hpj486" xml:space="preserve">{{TFintro}}

==Learning Project Introduction==
Welcome to Trade Finance. The objectives of this learning project are to help you:
*
*Evaluate financial risks and methods.
*Select and implement most favorable methods of payment to support global activities and ensure that all related costs are included at the time of quotation. 
*Evaluate quantity and source of finance necessary to implement global activities.

==Resource Sections==
*[[Trade Finance/Political and Economic Risk|Module 1: Political and Economic Risk]]
*[[Trade Finance/Risk Mitigation Techniques|Module 2: Risk Mitigation Techniques]]
*[[Trade Finance/Commercial Risk|Module 3: Commercial Risk]]
*[[Trade Finance/Payment Methods|Module 4: Payment Methods]]
*[[Trade Finance/Selecting Payment Methods|Module 5: Selecting Payment Methods]]
*[[Trade Finance/Financial Plan|Module 6: Financial Plan]]
*[[Trade Finance/Short-term Financing|Module 7: Short-term Financing]]
*[[Trade Finance/Medium- and Long-term Financing|Module 8: Medium- and Long-term Financing]]


==Resource References==
*[http://ibcertification.msu.edu International Business and the Certified Global Business Professional] (MSU Global)
*[http://www.nasbitecgbp.org NASBITE CGBP] (National Association of Small Business International Trade Educators- Certified Global Business Professional)


==About this Resource==
These resources were developed by MSU Global with funding provided by a  [http://www.ed.gov/programs/iegpsbie/index.html U.S. Department of Education, Business in International Education] Title VIB grant.

==See also==
{{tertiary}}
{{complete}}
{{finance}}
{{course}}
{{featured}}
* [[School:Business]]
* [[Portal:Finance]]

{{hitcounter}}

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      <text bytes="3202" sha1="39991u8tnof581gnczrnqvktmv2qdl8" xml:space="preserve">'''Renewable energy''' is energy that can be continuously be used and refilled drawing usually from natural regenerative sources or natural environmental processes such as wind, solar, and bio. It is often contrasted with non-renewable sources such as coal and fossil fuels which take longer to generate. Renewable energy uses energy flows that are replenished at the same rate as they are used. Renewable energy can potentially help to mitigate and/or alleviate various [[environmental problems]] potentially. 
{{RightTOC}}
About 13 percent of [[primary energy]] comes from renewables, with most of this coming from traditional biomass like [[Wood fuel|wood-burning]].  Hydropower is the next largest source, providing 2-3%, and modern technologies like geothermal, wind, solar, and marine energy together produce less than 1% of total world energy demand but are expanding rapidly.&lt;ref name=&quot;IEA&quot;&gt;International Energy Agency (2007). 
[http://www.iea.org/textbase/papers/2006/renewable_factsheet.pdf ''Renewables in global energy supply: An IEA facts sheet''], OECD, p. 3.&lt;/ref&gt;  The technical potential for their use is very large, exceeding all other readily available sources.&lt;ref&gt;World Energy Assessment (2001). [http://www.undp.org/energy/activities/wea/drafts-frame.html Renewable energy technologies], chapter 7.&lt;/ref&gt;

== Courses ==
Current renewable energy courses in progress include:

* [[Introduction to solar energy]]

== Examples ==
[[File:Germany Wind Turbines (15342713191).jpg|thumb|right|240px]]
Current renewable energy technologies include:
* Solar Energy
** [[/Solar thermal/]]
** [[/Photovoltaics/]]
** [[/Bioenergy/]]
** [[/Wind energy/]]
** [[/Hydroelectric/]]
* Non Solar Renewables
** [[/Tidal energy/]]
** [[/Geothermal energy/]]  

Wind:
* [[/Windstalk/]]

== Lessons ==
# [[Renewable energy/The energy challenge|The energy challenge]]
# [[Renewable energy/Semiconductors|Key concepts of semiconductors]]
# [[Renewable energy/Photovoltaics|Photovoltaics]]

== Additional resources ==
===Wikipedia===
* [[w:Hybrid renewable energy system|Hybrid renewable energy system]]
* [[w:Biogasoline|Biogasoline]]
* [[w:List of renewable energy topics by country and territory|List of renewable energy topics by country and territory]]
* [[w:List of renewable energy organizations|List of renewable energy organizations]]

===External resources===
* [http://en.openei.org OpenEI: Energy Data, Tools, Models, and other Resources]
* [http://www.tfot.info/content/view/124/71/ Open-source Renewable Energy Project]
* [[Appropedia:Renewable energy wiki|Overview of renewable energy wiki resources]] (Appropedia)
* USA: [http://www.ncat.org/ National Center for Appropriate Technology (NCAT)] | [http://www.nrel.gov/ National Renewable Energy Laboratory] | [http://www.eere.energy.gov/ U.S. DOE Energy Efficiency and Renewable Energy (EERE)]

== See also ==
* [[DIY home electricity system]] 
* [[Green building]] 
* [[MIT Energy Initiative]]
* [[Renewable energy systems]]
* [[Simplified DIY Rooftop Wind Turbine]]

== References ==
&lt;references/&gt;

==External links==
*[https://opensustain.tech/ Directory of open renewable technology]

[[Category:Ecological sustainability]]

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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>Atcovi moved page [[Ship strength]] to [[Naval architecture/Ship strength]]: moving under project</comment>
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      <text bytes="5490" sha1="aexqeu5t5955dfuxatzvhd70pe3wdj4" xml:space="preserve">The subject of Ship Strength deals with the assessment of the ship's structural design to withstand the service loads she will confront during her lifetime. It is of profound importance for the design and usability of the ship. During the design phase of the ship, the naval architect has to account for the type and service conditions of the ship under consideration in order to engineer its structural arrangement accordingly.

As is the case in many applications, the engineer has to compromise conflicting requirements. For seagoing ships, this conflict occurs usually between service requirements, targeted service conditions, ship life cycle and structural strength. For example, a crude oil tanker has to have the least structural weight (increase [[DWT]]) in order to increase profitability, sail under any conditions, survive on standard preventive maintenance for 30 years and yet have a structure that can deliver this. For a coast liner on the other hand, requirements may include among others high speed and increased passenger safety. 
{{RightTOC}}
Additionally, the structural arrangement has to provide for convenience and usability of the ship. For example, the compartmentation of the cargo and ballast area of the ship has to comply with strength criteria, pump/piping availability, ship stability and operations requirements. For example a tanker might have to receive and deliver particular quantities of oil. Thus the grouping of the cargo and ballast tanks along with the piping and pump arrangements should be such as to make it more convenient to unload such quantities as fast as possible. The remaining quantities however, will have to fulfill both stability and strength criteria. From the strength point of view, non-uniform loading conditions might result in complex loads on the structure, including transverse shear, bending and torsion. Moving cargo around is not a choice, as this will at least blow up the operations' scheduling: having to pump cargo out of all tanks takes much more time in the harbor and as usual time is money. But even more important is that half-filled tanks, may result in excessive [[sloshing]], which in turn can pose a considerable safety thread.

Apart from this &quot;high level&quot; structural design aspects, the structural arrangement has to provide for other usability issues that are more local. A ship comprises a constellation of different pieces of machinery: engines, power units, turbines, pumps, cranes, derricks, mooring equipment and so on. The structural arrangement must consider all these local loads and foresee for adequate foundation and support. Though this may seem it can be handled at the local level, it is not always so straight-forward: consider that a lot of the machinery lies on the main deck or even higher ones that are usually of smaller rigidity than the inner bottom for example.

The purpose of the above discussion is merely to illustrate few issues the Naval Architect has to face when confronted with the task of designing the structure of a ship. Consider that these are trivial issues: special types of craft might - and probably will - need much more elaboration at the design stage. However this introduction also outlines the different levels of structural design that the designer has to consider. Ships, and in particular large ones, are hollow structures composed of very small elements. Consider the [[midship section]] of a large [[double hull]] tanker: it might be 40 m in breadth and 20 m in depth. However, the ''actual'' area of the elements comprising the midship section, might be less than &lt;math&gt;5 m^2&lt;/math&gt;! That is, if you could squeeze it so that there are no gaps left, it wouldn't be larger than your carpet. The challenge is clear: design one of the biggest structures ever made by human beings.

== Load types exerted on marine structures ==
There are different ways to classify the loads that are exerted on a marine structure. As a first approach, a ship is considered along with all its equipment, cargo and fluids as the system under consideration. The loads exerted to this system could be classified into the following two categories:

* Standard Loads
Standard or operational loads are the ones that the ship will experience during most of her lifetime. These loads act to the ship as a whole as concentrated loads. Such loads include:
** Static still water and wave buoyancy: these are the hydrostatic forces that act to the ship hull when the ship is afloat.
** Dynamic lift loads: for semi-displacement and planning hulls.
** Wind pressure: especially for ships with large superstructure area.
** Drydock loads: when a ship lays on the drydock platform.
** Mooring lines and anchors: these act as concentrated loads.

* Extreme Loads
This class of loads occurs when the ship sails in harsh weather conditions. The naval architect should keep in mind that these loads may occur rarely. But they will occur and therefore they should never be neglected. Such loads include:
** Ship-Ship and ship-obstacle collisions.
** Grounding.
** Slamming.
** Green sea.

== Ship Equilibrium ==

== Ship Structural Arrangement ==

== Ship floating on still water ==

== Ship in waves ==

== Normal Stresses ==

== Shear Stresses ==

== Torsion ==

[[Category:Marine engineering]]

==Active participants==
Active participants in this [[Portal:Learning Projects#Learning Groups|Learning Group]]
* ...


[[Category:Naval architecture|Naval Architecture]]
[[Category:Ships|strength]]</text>
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  </page>
  <page>
    <title>Energy stored by a capacitor</title>
    <ns>0</ns>
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      <id>2810589</id>
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      <timestamp>2026-05-20T12:39:10Z</timestamp>
      <contributor>
        <username>Atcovi</username>
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      <minor />
      <comment>project box(es)</comment>
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      <text bytes="1369" sha1="j1i5v0jsu3ven3k4b290wasifrlpztj" xml:space="preserve">{{engineering}}
{{physics}}
The [[energy]] (measured in [[Joule]]s) stored in a [[Electronics/Capacitors|capacitor]] is equal to the ''work'' done to charge it. Consider a capacitance ''C'', holding a charge ''+q'' on one plate and ''-q'' on the other. Moving a small element of charge &lt;math&gt;\mathrm{d}q&lt;/math&gt; from one plate to the other against the potential difference ''V = q/C'' requires the work &lt;math&gt;\mathrm{d}W&lt;/math&gt;:
   
:&lt;math&gt; \mathrm{d}W = \frac{q}{C}\,\mathrm{d}q &lt;/math&gt;

where

:''W'' is the work measured in joules

:''q'' is the charge measured in coulombs

:''C'' is the capacitance, measured in farads

We can find the energy stored in a capacitance by [[Foundations_of_Calculus|integrating]] this equation. Starting with an uncharged capacitance (''q''=0) and moving charge from one plate to the other until the plates have charge ''+Q'' and ''-Q'' requires the work ''W'':

:&lt;math&gt; W_{charging} = \int_{0}^{Q} \frac{q}{C} \, \mathrm{d}q = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}CV^2 = W_{stored}&lt;/math&gt;

Combining this with the above equation for the capacitance of a flat-plate capacitor, we get:

:&lt;math&gt; W_{stored} = \frac{1}{2} C V^2 = \frac{1}{2} \epsilon \frac{A}{d} V^2&lt;/math&gt; .

where

:''W'' is the energy measured in joules

:''C'' is the capacitance, measured in farads

:''V'' is the voltage measured in volts

[[Category:Electronics]]</text>
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  <page>
    <title>Rwandan Genocide</title>
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      <id>2810601</id>
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      <timestamp>2026-05-20T14:14:58Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>project box(es)</comment>
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      <text bytes="1835" sha1="teb9nw00qrilxoqwpjvrd50vamwd8nc" xml:space="preserve">{{history}}
{{history-stub}}

The '''Rwandan Genocide''' (1994) involved the mass killing of hundreds of thousands of ethnic [[w:Tutsi|Tutsi]]s and moderate '''[[w:Hutu|Hutu]]''' sympathizers in [[Rwanda]] and was the largest atrocity during the [[w:Rwandan Civil War|Rwandan Civil War]] and was one of the most significant human atrocities of the twentieth century. 

==Overview==
[[Image:Kigali Memorial Centre 5.jpg|275px|thumb|right|Kigali Memorial Centre, Gisozi, Rwanda.]]
* The Rwandan [[genocide]] was mostly carried out by two extremist [[w:Hutu|Hutu]] militia groups, the [[w:Interahamwe|Interahamwe]] and the [[w:Impuzamugambi|Impuzamugambi]], during about 100 days from April 6 through mid-July, 1994.
* Somewhere between half a million and one million people were systematically slaughtered in three months (~100 days).
* Approx. 14% of the [[Rwanda|Rwandan]] population died, including approximately 77% of [[w:Tutsi|Tutsis]] living in Rwanda.
* This is numerically equivalent to approximately three [[w:September 11, 2001 attacks|September 11]] incidents occurring every day for three months.
* At least 500,000 [[w:Tutsi|Tutsis]] and thousands of moderate [[w:Hutus|Hutus]] died. Note that several estimates put the death toll between 700,000 and 1,000,000, depending on the source.

==See also==
* [[w:Bibliography of the Rwandan genocide|Bibliography of the Rwandan genocide]]
* [[w:Gacaca|Gacaca courts]] - the community justice system used after the 1994 genocide
* [[Ghosts of Rwanda]] (a documentary)

{{wikipedia}}
{{commons}}

==External links==
* [http://news.bbc.co.uk/2/hi/africa/3580247.stm Timeline: 100 days of genocide] (BBC News, 6 April, 2004)
* [http://www.pbs.org/wgbh/pages/frontline/shows/evil/ The triumph of evil] (pbs.org, interviews, time line, further readings) 

[[Category:Rwandan Genocide]]</text>
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  <page>
    <title>Naval architecture/Ship Construction</title>
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        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>Atcovi moved page [[Ship Construction]] to [[Naval architecture/Ship Construction]]: moving under project</comment>
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      <text bytes="3085" sha1="1njt1c5jagpjhgo9oy5qmv8w91j4key" xml:space="preserve">Stages of Ship Construction

1.) Gathering, mobilization and sorting of materials
Materials are arranged in such a way that the last that was being filed up on top is the first material to be cut. It is sorted according to thickness plate, size of bulb flats, angle bars, etc.

2.) Cutting of materials – now a day’s materials are cut by [[w:Plasma cutting|CNC machining]], draftsman from the Engineering or Design section of the Shipyard will provide stl or jpeg file drawings, they saved it in a floppy disk or portable USD flash disk. This is being inserted on the CNC machine computer control board then the machine will cut according to shape of the detailed drawing.

3.) Fabrication will start from the basic component of the Ship “the Panel”, during this stage this is called the “Panel wise fabrication”. Plate is being joined together in a horizontal level ground for straight panels, pin jigs are being used to support curved panels wherein not all surface is tangent to the level ground. Plate is being joined by {{w|submerged arc welding}}. There are horizontal or vertical members that are being fitting during this stage; these are the girders and stiffeners. Girders are the rigid frame of the ship whiles stiffeners are the sub-members in between girders or frame. During this stage panels are fabricated in either of the following part of the ship; main deck transverse bulkhead, longitudinal bulkhead, side shell, inner bottom or tank top and bottom shell.

4.) Panels are now joined together to form a block inside a graving dock or floating dock – a ship is subdivided according to several blocks in either of the following blocks; fore peak tank, side ballast tanks, center cargo hold compartment, engine room, after peak tank and accommodation block. Blocks are joined by welding a ceramic filled welding in an FCAW process ([[w:FCAW|Flux Cored Arc Welding]]). Each and every stage of Ship Construction there are frequent inspection of a trained Quality Assurance personnel that are knowledgeable in Steel fabrication, welding, NDT and well verse to the standard criteria of Classification Society like IACS ({{w|International Association of Classification Societies}}) Inspection records are maintained through all stages in each fitter’s traceability, welder’s traceability, dimension traceability, NDT ([[w:Nondestructive testing|non-destructive testing]]) traceability and material traceability.

5.) Careful planning and execution of schedules are made in each stages to ensure that there’s no component that are left behind, we cannot cut the ship’s compartment again to install the left behind valves, pipe parts, pumps, etc.

6.) There are several departments involved in Ship building; Hull – which is the structural department, pipe, outfitting - for the non-structural components of hull such as manhole, hatch cover, bobby hatch, railings, tank supports etc. machinery department for auxiliary engine, main engine, reduction gear, governor, heat exchanger, etc.

[[Category:Naval architecture]]
[[Category:Ships|Construction]]</text>
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{{title|Motivation and emotion image gallery}}
* This gallery features images related to [[motivation and emotion]] from [[commons:|Wikimedia Commons]]
* These images may be useful for embedding in [[Motivation and emotion/Book|motivation and emotion book chapter]]s to help illustrate key points and to provide examples
* To find other images, search [[commons:|Commons]]. You can also contribute new media to Commons. 
* For more info about selecting, embedding, and managing images for this project, see [[Motivation and emotion/Wikiversity/Figures|working with figures]]
&lt;/noinclude&gt;
{{center top}}
&lt;gallery&gt;
File:1944 JonWhitcomb USNavy (3214638694).jpg
File:2010 - A year plenty of Hopes.jpg
File:Adam and Eve (UK CIA P-1947-LF-77).jpg
File:AG LEADER.JPG
File:Alienation.jpg
File:Alma-Tadema Unconscious Rivals 1893.jpg
File:Amarguraubeda.JPG
File:Angelo Bronzino 003.jpg
File:Arianna e la sua lente.JPG
File:B&amp;W Happiness.jpg
File:BB-Bea.jpg
File:Bipolar Dyptych 1 365.jpg
File:Bundesarchiv Bild 183-1984-0809-003, Kyffhäuserhütte Artern, Jugendforscherkollektiv.jpg
File:Child's Angry Face.jpg
File:Concert exercise.jpg
File:Contempt.jpg
File:Crying child with blonde hair.jpg
File:Cycling Time Trial effort.jpg
File:De mulieribus claris - Marcia.png
File:Depression.jpg
File:Disappointment facial expression.jpg
File:Disgust expression cropped.jpg
File:Disgust1.jpg
File:Drill sergeant screams.jpg
File:Doctor Who (13).jpg
File:Edward Lear A Book of Nonsense 57.jpg
File:Eeg registration.jpg
File:El Lehendakari visita la nueva unidad de Resonancia Magnética del HUA (40116421582).jpg
File:Emo boy 03 in rage.jpg
File:Emotions wheel.png
File:Empathy and the brain.png
File:Empathy facial expression 2.png
File:Eros bobbin Louvre CA1798.jpg
File:Expression of the Emotions Figure 1.png
File:Expression of the Emotions Figure 20.png
File:Faradarmani.gif
File:Forestay-Eye-Round-seizings-Bulls-eye.jpg
File:Folsom Hula Hoop.png
File:Fragile Emotion cropped.jpg
File:Georg Friedrich Kersting Kinder am Fenster.jpg
File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG
File:Grass of Happiness.jpg
File:Gustave Courbet - Le Désespéré (1843).jpg
File:Inside my head.jpg
File:Interest.jpg
File:It's a Braun definitely.jpg
File:Just love cropped.jpg
File:Little girl buried up to her neck in sand.jpg
File:Lotus Reflects the Sun.jpg
File:Love Rush!.jpg
File:Love's Passing.jpg
File:Man eats at Volunteers of America soup kitchen Washington DC 1936.gif
File:Mary Magdalene Crying Statue.jpg
File:Mascia Ferri in Speed Skydiving.jpg
File:Mood dice.svg
File:Motivation &amp; Emotion.png
File:Motivation and Emotion Scrabble.jpg
File:Motivation Laptop.svg
File:Museum für Ostasiatische Kunst Dahlem Berlin Mai 2006 029.jpg
File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg
File:Noun emotion 1325508.svg
File:One hand handstand.jpg
File:Paris - Playing chess at the Jardins du Luxembourg - 2966.jpg
File:Person holding clock in front of head.jpg
File:Picswiss_UR-25-03.jpg
File:Plato Pio-Clemetino Inv305.jpg
File:Plutchik-wheel.svg
File:Portrait gemma and mehmet.jpg
File:Rock climbing (B&amp;W).jpg
File:Sadness 2.jpg
File:Sadness.jpg
File:Schadenfreude.png
File:Sépulcre Arc-en-Barrois 111008 12.jpg
File:Sigmund Freud LIFE.jpg
File:Smiling Brazilian girl (black and white).jpg
File:Smiling girl.jpg
File:Sophia (robot).jpg
File:Souvenir Seller - Moscow - Russia.JPG
File:SURPRISE.jpg
File:Thorunn3.jpg
File:Tug-of-war.jpg
File:Two people laughing.jpg
File:Two people with brain cogs turning.png
File:US Navy 100810-N-3013W-014 A Drug Education for Youth (DEFY) summer camp attendee from Naval Air Station Jacksonville climbs a rock wall during a goal setting exercise at Camp McConnell in Micanopy, Fla.jpg
File:Vault figure.jpg
File:Vincent Van Gogh - Sorrow.JPG
File:WebKit logo.svg
File:Week-end pleasure.jpg
File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg
File:Worried little girl.jpg
File:Yakunchikova Fear.jpg
File:You may now kiss the bride.jpg
&lt;/gallery&gt;
{{center bottom}}
{{center top}}
For more images, search &lt;span class=&quot;plainlinks&quot;&gt;[http://commons.wikimedia.org/w/index.php?title=Special:Search&amp;fulltext=Search&amp;ns0=1&amp;ns1=1&amp;ns2=1&amp;ns3=1&amp;ns4=1&amp;ns5=1&amp;ns6=1&amp;ns7=1&amp;ns8=1&amp;ns9=1&amp;ns10=1&amp;ns11=1&amp;ns12=1&amp;ns13=1&amp;ns14=1&amp;ns15=1&amp;ns100=1&amp;ns101=1&amp;ns102=1&amp;ns103=1&amp;redirs=0&amp;search= wiki commons].&lt;/span&gt;
{{center bottom}}
&lt;noinclude&gt;
[[Category:Motivation and emotion]]
[[Category:Galleries]]
&lt;/noinclude&gt;</text>
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      <comment>+  File:Man with superimposed brain.jpg</comment>
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{{title|Motivation and emotion image gallery}}
* This gallery features images related to [[motivation and emotion]] from [[commons:|Wikimedia Commons]]
* These images may be useful for embedding in [[Motivation and emotion/Book|motivation and emotion book chapter]]s to help illustrate key points and to provide examples
* To find other images, search [[commons:|Commons]]. You can also contribute new media to Commons. 
* For more info about selecting, embedding, and managing images for this project, see [[Motivation and emotion/Wikiversity/Figures|working with figures]]
&lt;/noinclude&gt;
{{center top}}
&lt;gallery&gt;
File:1944 JonWhitcomb USNavy (3214638694).jpg
File:2010 - A year plenty of Hopes.jpg
File:Adam and Eve (UK CIA P-1947-LF-77).jpg
File:AG LEADER.JPG
File:Alienation.jpg
File:Alma-Tadema Unconscious Rivals 1893.jpg
File:Amarguraubeda.JPG
File:Angelo Bronzino 003.jpg
File:Arianna e la sua lente.JPG
File:B&amp;W Happiness.jpg
File:BB-Bea.jpg
File:Bipolar Dyptych 1 365.jpg
File:Bundesarchiv Bild 183-1984-0809-003, Kyffhäuserhütte Artern, Jugendforscherkollektiv.jpg
File:Child's Angry Face.jpg
File:Concert exercise.jpg
File:Contempt.jpg
File:Crying child with blonde hair.jpg
File:Cycling Time Trial effort.jpg
File:De mulieribus claris - Marcia.png
File:Depression.jpg
File:Disappointment facial expression.jpg
File:Disgust expression cropped.jpg
File:Disgust1.jpg
File:Drill sergeant screams.jpg
File:Doctor Who (13).jpg
File:Edward Lear A Book of Nonsense 57.jpg
File:Eeg registration.jpg
File:El Lehendakari visita la nueva unidad de Resonancia Magnética del HUA (40116421582).jpg
File:Emo boy 03 in rage.jpg
File:Emotions wheel.png
File:Empathy and the brain.png
File:Empathy facial expression 2.png
File:Eros bobbin Louvre CA1798.jpg
File:Expression of the Emotions Figure 1.png
File:Expression of the Emotions Figure 20.png
File:Faradarmani.gif
File:Forestay-Eye-Round-seizings-Bulls-eye.jpg
File:Folsom Hula Hoop.png
File:Fragile Emotion cropped.jpg
File:Georg Friedrich Kersting Kinder am Fenster.jpg
File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG
File:Grass of Happiness.jpg
File:Gustave Courbet - Le Désespéré (1843).jpg
File:Inside my head.jpg
File:Interest.jpg
File:It's a Braun definitely.jpg
File:Just love cropped.jpg
File:Little girl buried up to her neck in sand.jpg
File:Lotus Reflects the Sun.jpg
File:Love Rush!.jpg
File:Love's Passing.jpg
File:Man eats at Volunteers of America soup kitchen Washington DC 1936.gif
File:Man with superimposed brain.jpg
File:Mary Magdalene Crying Statue.jpg
File:Mascia Ferri in Speed Skydiving.jpg
File:Mood dice.svg
File:Motivation &amp; Emotion.png
File:Motivation and Emotion Scrabble.jpg
File:Motivation Laptop.svg
File:Museum für Ostasiatische Kunst Dahlem Berlin Mai 2006 029.jpg
File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg
File:Noun emotion 1325508.svg
File:One hand handstand.jpg
File:Paris - Playing chess at the Jardins du Luxembourg - 2966.jpg
File:Person holding clock in front of head.jpg
File:Picswiss_UR-25-03.jpg
File:Plato Pio-Clemetino Inv305.jpg
File:Plutchik-wheel.svg
File:Portrait gemma and mehmet.jpg
File:Rock climbing (B&amp;W).jpg
File:Sadness 2.jpg
File:Sadness.jpg
File:Schadenfreude.png
File:Sépulcre Arc-en-Barrois 111008 12.jpg
File:Sigmund Freud LIFE.jpg
File:Smiling Brazilian girl (black and white).jpg
File:Smiling girl.jpg
File:Sophia (robot).jpg
File:Souvenir Seller - Moscow - Russia.JPG
File:SURPRISE.jpg
File:Thorunn3.jpg
File:Tug-of-war.jpg
File:Two people laughing.jpg
File:Two people with brain cogs turning.png
File:US Navy 100810-N-3013W-014 A Drug Education for Youth (DEFY) summer camp attendee from Naval Air Station Jacksonville climbs a rock wall during a goal setting exercise at Camp McConnell in Micanopy, Fla.jpg
File:Vault figure.jpg
File:Vincent Van Gogh - Sorrow.JPG
File:WebKit logo.svg
File:Week-end pleasure.jpg
File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg
File:Worried little girl.jpg
File:Yakunchikova Fear.jpg
File:You may now kiss the bride.jpg
&lt;/gallery&gt;
{{center bottom}}
{{center top}}
For more images, search &lt;span class=&quot;plainlinks&quot;&gt;[http://commons.wikimedia.org/w/index.php?title=Special:Search&amp;fulltext=Search&amp;ns0=1&amp;ns1=1&amp;ns2=1&amp;ns3=1&amp;ns4=1&amp;ns5=1&amp;ns6=1&amp;ns7=1&amp;ns8=1&amp;ns9=1&amp;ns10=1&amp;ns11=1&amp;ns12=1&amp;ns13=1&amp;ns14=1&amp;ns15=1&amp;ns100=1&amp;ns101=1&amp;ns102=1&amp;ns103=1&amp;redirs=0&amp;search= wiki commons].&lt;/span&gt;
{{center bottom}}
&lt;noinclude&gt;
[[Category:Motivation and emotion]]
[[Category:Galleries]]
&lt;/noinclude&gt;</text>
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{{title|Motivation and emotion image gallery}}
* This gallery features images related to [[motivation and emotion]] from [[commons:|Wikimedia Commons]]
* These images may be useful for embedding in [[Motivation and emotion/Book|motivation and emotion book chapter]]s to help illustrate key points and to provide examples
* To find other images, search [[commons:|Commons]]. You can also contribute new media to Commons. 
* For more info about selecting, embedding, and managing images for this project, see [[Motivation and emotion/Wikiversity/Figures|working with figures]]
&lt;/noinclude&gt;
{{center top}}
&lt;gallery&gt;
File:1944 JonWhitcomb USNavy (3214638694).jpg
File:2010 - A year plenty of Hopes.jpg
File:Adam and Eve (UK CIA P-1947-LF-77).jpg
File:AG LEADER.JPG
File:Alienation.jpg
File:Alma-Tadema Unconscious Rivals 1893.jpg
File:Amarguraubeda.JPG
File:Angelo Bronzino 003.jpg
File:Arianna e la sua lente.JPG
File:B&amp;W Happiness.jpg
File:BB-Bea.jpg
File:Bipolar Dyptych 1 365.jpg
File:Bundesarchiv Bild 183-1984-0809-003, Kyffhäuserhütte Artern, Jugendforscherkollektiv.jpg
File:Child's Angry Face.jpg
File:Concert exercise.jpg
File:Contempt.jpg
File:Crying child with blonde hair.jpg
File:Cycling Time Trial effort.jpg
File:DALL-E Generative AI Brain VanGogh.png
File:De mulieribus claris - Marcia.png
File:Depression.jpg
File:Disappointment facial expression.jpg
File:Disgust expression cropped.jpg
File:Disgust1.jpg
File:Drill sergeant screams.jpg
File:Doctor Who (13).jpg
File:Edward Lear A Book of Nonsense 57.jpg
File:Eeg registration.jpg
File:El Lehendakari visita la nueva unidad de Resonancia Magnética del HUA (40116421582).jpg
File:Emo boy 03 in rage.jpg
File:Emotions wheel.png
File:Empathy and the brain.png
File:Empathy facial expression 2.png
File:Eros bobbin Louvre CA1798.jpg
File:Expression of the Emotions Figure 1.png
File:Expression of the Emotions Figure 20.png
File:Faradarmani.gif
File:Forestay-Eye-Round-seizings-Bulls-eye.jpg
File:Folsom Hula Hoop.png
File:Fragile Emotion cropped.jpg
File:Georg Friedrich Kersting Kinder am Fenster.jpg
File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG
File:Grass of Happiness.jpg
File:Gustave Courbet - Le Désespéré (1843).jpg
File:Inside my head.jpg
File:Interest.jpg
File:It's a Braun definitely.jpg
File:Just love cropped.jpg
File:Little girl buried up to her neck in sand.jpg
File:Lotus Reflects the Sun.jpg
File:Love Rush!.jpg
File:Love's Passing.jpg
File:Man eats at Volunteers of America soup kitchen Washington DC 1936.gif
File:Man with superimposed brain.jpg
File:Mary Magdalene Crying Statue.jpg
File:Mascia Ferri in Speed Skydiving.jpg
File:Mood dice.svg
File:Motivation &amp; Emotion.png
File:Motivation and Emotion Scrabble.jpg
File:Motivation Laptop.svg
File:Museum für Ostasiatische Kunst Dahlem Berlin Mai 2006 029.jpg
File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg
File:Noun emotion 1325508.svg
File:One hand handstand.jpg
File:Paris - Playing chess at the Jardins du Luxembourg - 2966.jpg
File:Person holding clock in front of head.jpg
File:Picswiss_UR-25-03.jpg
File:Plato Pio-Clemetino Inv305.jpg
File:Plutchik-wheel.svg
File:Portrait gemma and mehmet.jpg
File:Rock climbing (B&amp;W).jpg
File:Sadness 2.jpg
File:Sadness.jpg
File:Schadenfreude.png
File:Sépulcre Arc-en-Barrois 111008 12.jpg
File:Sigmund Freud LIFE.jpg
File:Smiling Brazilian girl (black and white).jpg
File:Smiling girl.jpg
File:Sophia (robot).jpg
File:Souvenir Seller - Moscow - Russia.JPG
File:SURPRISE.jpg
File:Thorunn3.jpg
File:Tug-of-war.jpg
File:Two people laughing.jpg
File:Two people with brain cogs turning.png
File:US Navy 100810-N-3013W-014 A Drug Education for Youth (DEFY) summer camp attendee from Naval Air Station Jacksonville climbs a rock wall during a goal setting exercise at Camp McConnell in Micanopy, Fla.jpg
File:Vault figure.jpg
File:Vincent Van Gogh - Sorrow.JPG
File:WebKit logo.svg
File:Week-end pleasure.jpg
File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg
File:Worried little girl.jpg
File:Yakunchikova Fear.jpg
File:You may now kiss the bride.jpg
&lt;/gallery&gt;
{{center bottom}}
{{center top}}
For more images, search &lt;span class=&quot;plainlinks&quot;&gt;[http://commons.wikimedia.org/w/index.php?title=Special:Search&amp;fulltext=Search&amp;ns0=1&amp;ns1=1&amp;ns2=1&amp;ns3=1&amp;ns4=1&amp;ns5=1&amp;ns6=1&amp;ns7=1&amp;ns8=1&amp;ns9=1&amp;ns10=1&amp;ns11=1&amp;ns12=1&amp;ns13=1&amp;ns14=1&amp;ns15=1&amp;ns100=1&amp;ns101=1&amp;ns102=1&amp;ns103=1&amp;redirs=0&amp;search= wiki commons].&lt;/span&gt;
{{center bottom}}
&lt;noinclude&gt;
[[Category:Motivation and emotion]]
[[Category:Galleries]]
&lt;/noinclude&gt;</text>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Consolidate text</comment>
      <origin>2810732</origin>
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      <text bytes="4534" sha1="1jnrb0zxab936yiajewlojtwpun5nfa" xml:space="preserve">&lt;noinclude&gt;
{{title|Motivation and emotion image gallery}}
* This gallery features images related to [[motivation and emotion]] from [[commons:|Wikimedia Commons]].
* These images may be useful for embedding in [[Motivation and emotion/Book|motivation and emotion book chapter]]s to help illustrate key points and to provide examples.
* To find other images, &lt;span class=&quot;plainlinks&quot;&gt;[http://commons.wikimedia.org/w/index.php?title=Special:Search&amp;fulltext=Search&amp;ns0=1&amp;ns1=1&amp;ns2=1&amp;ns3=1&amp;ns4=1&amp;ns5=1&amp;ns6=1&amp;ns7=1&amp;ns8=1&amp;ns9=1&amp;ns10=1&amp;ns11=1&amp;ns12=1&amp;ns13=1&amp;ns14=1&amp;ns15=1&amp;ns100=1&amp;ns101=1&amp;ns102=1&amp;ns103=1&amp;redirs=0&amp;search= search Wikimedia Commons].&lt;/span&gt;
. You can also contribute new media to Commons. 
* For more info about selecting, embedding, and managing images for this project, see [[Motivation and emotion/Wikiversity/Figures|working with figures]].&lt;/noinclude&gt;
{{center top}}
&lt;gallery&gt;
File:1944 JonWhitcomb USNavy (3214638694).jpg
File:2010 - A year plenty of Hopes.jpg
File:Adam and Eve (UK CIA P-1947-LF-77).jpg
File:AG LEADER.JPG
File:Alienation.jpg
File:Alma-Tadema Unconscious Rivals 1893.jpg
File:Amarguraubeda.JPG
File:Angelo Bronzino 003.jpg
File:Arianna e la sua lente.JPG
File:B&amp;W Happiness.jpg
File:BB-Bea.jpg
File:Bipolar Dyptych 1 365.jpg
File:Bundesarchiv Bild 183-1984-0809-003, Kyffhäuserhütte Artern, Jugendforscherkollektiv.jpg
File:Child's Angry Face.jpg
File:Concert exercise.jpg
File:Contempt.jpg
File:Crying child with blonde hair.jpg
File:Cycling Time Trial effort.jpg
File:DALL-E Generative AI Brain VanGogh.png
File:De mulieribus claris - Marcia.png
File:Depression.jpg
File:Disappointment facial expression.jpg
File:Disgust expression cropped.jpg
File:Disgust1.jpg
File:Drill sergeant screams.jpg
File:Doctor Who (13).jpg
File:Edward Lear A Book of Nonsense 57.jpg
File:Eeg registration.jpg
File:El Lehendakari visita la nueva unidad de Resonancia Magnética del HUA (40116421582).jpg
File:Emo boy 03 in rage.jpg
File:Emotions wheel.png
File:Empathy and the brain.png
File:Empathy facial expression 2.png
File:Eros bobbin Louvre CA1798.jpg
File:Expression of the Emotions Figure 1.png
File:Expression of the Emotions Figure 20.png
File:Faradarmani.gif
File:Forestay-Eye-Round-seizings-Bulls-eye.jpg
File:Folsom Hula Hoop.png
File:Fragile Emotion cropped.jpg
File:Georg Friedrich Kersting Kinder am Fenster.jpg
File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG
File:Grass of Happiness.jpg
File:Gustave Courbet - Le Désespéré (1843).jpg
File:Inside my head.jpg
File:Interest.jpg
File:It's a Braun definitely.jpg
File:Just love cropped.jpg
File:Little girl buried up to her neck in sand.jpg
File:Lotus Reflects the Sun.jpg
File:Love Rush!.jpg
File:Love's Passing.jpg
File:Man eats at Volunteers of America soup kitchen Washington DC 1936.gif
File:Man with superimposed brain.jpg
File:Mary Magdalene Crying Statue.jpg
File:Mascia Ferri in Speed Skydiving.jpg
File:Mood dice.svg
File:Motivation &amp; Emotion.png
File:Motivation and Emotion Scrabble.jpg
File:Motivation Laptop.svg
File:Museum für Ostasiatische Kunst Dahlem Berlin Mai 2006 029.jpg
File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg
File:Noun emotion 1325508.svg
File:One hand handstand.jpg
File:Paris - Playing chess at the Jardins du Luxembourg - 2966.jpg
File:Person holding clock in front of head.jpg
File:Picswiss_UR-25-03.jpg
File:Plato Pio-Clemetino Inv305.jpg
File:Plutchik-wheel.svg
File:Portrait gemma and mehmet.jpg
File:Rock climbing (B&amp;W).jpg
File:Sadness 2.jpg
File:Sadness.jpg
File:Schadenfreude.png
File:Sépulcre Arc-en-Barrois 111008 12.jpg
File:Sigmund Freud LIFE.jpg
File:Smiling Brazilian girl (black and white).jpg
File:Smiling girl.jpg
File:Sophia (robot).jpg
File:Souvenir Seller - Moscow - Russia.JPG
File:SURPRISE.jpg
File:Thorunn3.jpg
File:Tug-of-war.jpg
File:Two people laughing.jpg
File:Two people with brain cogs turning.png
File:US Navy 100810-N-3013W-014 A Drug Education for Youth (DEFY) summer camp attendee from Naval Air Station Jacksonville climbs a rock wall during a goal setting exercise at Camp McConnell in Micanopy, Fla.jpg
File:Vault figure.jpg
File:Vincent Van Gogh - Sorrow.JPG
File:WebKit logo.svg
File:Week-end pleasure.jpg
File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg
File:Worried little girl.jpg
File:Yakunchikova Fear.jpg
File:You may now kiss the bride.jpg
&lt;/gallery&gt;
{{center bottom}}&lt;noinclude&gt;
[[Category:Motivation and emotion]]
[[Category:Galleries]]
&lt;/noinclude&gt;</text>
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    <revision>
      <id>2810733</id>
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      <timestamp>2026-05-21T04:54:04Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ upload link</comment>
      <origin>2810733</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4580" sha1="jxybvzvs8lnwsqtavcs3g5syjsfm2qe" xml:space="preserve">&lt;noinclude&gt;
{{title|Motivation and emotion image gallery}}
* This gallery features images related to [[motivation and emotion]] from [[commons:|Wikimedia Commons]].
* These images may be useful for embedding in [[Motivation and emotion/Book|motivation and emotion book chapter]]s to help illustrate key points and to provide examples.
* To find other images, &lt;span class=&quot;plainlinks&quot;&gt;[http://commons.wikimedia.org/w/index.php?title=Special:Search&amp;fulltext=Search&amp;ns0=1&amp;ns1=1&amp;ns2=1&amp;ns3=1&amp;ns4=1&amp;ns5=1&amp;ns6=1&amp;ns7=1&amp;ns8=1&amp;ns9=1&amp;ns10=1&amp;ns11=1&amp;ns12=1&amp;ns13=1&amp;ns14=1&amp;ns15=1&amp;ns100=1&amp;ns101=1&amp;ns102=1&amp;ns103=1&amp;redirs=0&amp;search= search Wikimedia Commons].&lt;/span&gt;
* You can also [[c:Special:UploadWizard|contribute new or freely licensed media to Commons]]. 
* For more info about selecting, embedding, and managing images for this project, see [[Motivation and emotion/Wikiversity/Figures|working with figures]].&lt;/noinclude&gt;
{{center top}}
&lt;gallery&gt;
File:1944 JonWhitcomb USNavy (3214638694).jpg
File:2010 - A year plenty of Hopes.jpg
File:Adam and Eve (UK CIA P-1947-LF-77).jpg
File:AG LEADER.JPG
File:Alienation.jpg
File:Alma-Tadema Unconscious Rivals 1893.jpg
File:Amarguraubeda.JPG
File:Angelo Bronzino 003.jpg
File:Arianna e la sua lente.JPG
File:B&amp;W Happiness.jpg
File:BB-Bea.jpg
File:Bipolar Dyptych 1 365.jpg
File:Bundesarchiv Bild 183-1984-0809-003, Kyffhäuserhütte Artern, Jugendforscherkollektiv.jpg
File:Child's Angry Face.jpg
File:Concert exercise.jpg
File:Contempt.jpg
File:Crying child with blonde hair.jpg
File:Cycling Time Trial effort.jpg
File:DALL-E Generative AI Brain VanGogh.png
File:De mulieribus claris - Marcia.png
File:Depression.jpg
File:Disappointment facial expression.jpg
File:Disgust expression cropped.jpg
File:Disgust1.jpg
File:Drill sergeant screams.jpg
File:Doctor Who (13).jpg
File:Edward Lear A Book of Nonsense 57.jpg
File:Eeg registration.jpg
File:El Lehendakari visita la nueva unidad de Resonancia Magnética del HUA (40116421582).jpg
File:Emo boy 03 in rage.jpg
File:Emotions wheel.png
File:Empathy and the brain.png
File:Empathy facial expression 2.png
File:Eros bobbin Louvre CA1798.jpg
File:Expression of the Emotions Figure 1.png
File:Expression of the Emotions Figure 20.png
File:Faradarmani.gif
File:Forestay-Eye-Round-seizings-Bulls-eye.jpg
File:Folsom Hula Hoop.png
File:Fragile Emotion cropped.jpg
File:Georg Friedrich Kersting Kinder am Fenster.jpg
File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG
File:Grass of Happiness.jpg
File:Gustave Courbet - Le Désespéré (1843).jpg
File:Inside my head.jpg
File:Interest.jpg
File:It's a Braun definitely.jpg
File:Just love cropped.jpg
File:Little girl buried up to her neck in sand.jpg
File:Lotus Reflects the Sun.jpg
File:Love Rush!.jpg
File:Love's Passing.jpg
File:Man eats at Volunteers of America soup kitchen Washington DC 1936.gif
File:Man with superimposed brain.jpg
File:Mary Magdalene Crying Statue.jpg
File:Mascia Ferri in Speed Skydiving.jpg
File:Mood dice.svg
File:Motivation &amp; Emotion.png
File:Motivation and Emotion Scrabble.jpg
File:Motivation Laptop.svg
File:Museum für Ostasiatische Kunst Dahlem Berlin Mai 2006 029.jpg
File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg
File:Noun emotion 1325508.svg
File:One hand handstand.jpg
File:Paris - Playing chess at the Jardins du Luxembourg - 2966.jpg
File:Person holding clock in front of head.jpg
File:Picswiss_UR-25-03.jpg
File:Plato Pio-Clemetino Inv305.jpg
File:Plutchik-wheel.svg
File:Portrait gemma and mehmet.jpg
File:Rock climbing (B&amp;W).jpg
File:Sadness 2.jpg
File:Sadness.jpg
File:Schadenfreude.png
File:Sépulcre Arc-en-Barrois 111008 12.jpg
File:Sigmund Freud LIFE.jpg
File:Smiling Brazilian girl (black and white).jpg
File:Smiling girl.jpg
File:Sophia (robot).jpg
File:Souvenir Seller - Moscow - Russia.JPG
File:SURPRISE.jpg
File:Thorunn3.jpg
File:Tug-of-war.jpg
File:Two people laughing.jpg
File:Two people with brain cogs turning.png
File:US Navy 100810-N-3013W-014 A Drug Education for Youth (DEFY) summer camp attendee from Naval Air Station Jacksonville climbs a rock wall during a goal setting exercise at Camp McConnell in Micanopy, Fla.jpg
File:Vault figure.jpg
File:Vincent Van Gogh - Sorrow.JPG
File:WebKit logo.svg
File:Week-end pleasure.jpg
File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg
File:Worried little girl.jpg
File:Yakunchikova Fear.jpg
File:You may now kiss the bride.jpg
&lt;/gallery&gt;
{{center bottom}}&lt;noinclude&gt;
[[Category:Motivation and emotion]]
[[Category:Galleries]]
&lt;/noinclude&gt;</text>
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  <page>
    <title>Motivation and emotion/Lectures/Brain and physiological needs</title>
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      <comment>Replace 2nd image with one that is more dynamically illustrative and revise the caption</comment>
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      <text bytes="8574" sha1="odhd4utdlvz9j1p4x8o1s5q0oyyab90" xml:space="preserve">{{Motivation and emotion/Lectures|Lecture 03: Brain and physiological needs|third}}
{{Motivation and emotion/Lectures/Complete}}
&lt;!-- {{Motivation and emotion/Lectures/In development}} --&gt;
&lt;!-- {{Motivation and emotion/Lectures/Complete}} --&gt;
[[File:WP20Symbols brain.svg|250px|right]]
==Overview==
This lecture:
* explains the role of [[Motivation and emotion/Brain structures|brain structures]], [[Motivation and emotion/Neurotransmitters|neurotransmitters]], and [[Motivation and emotion/Hormones|hormones]] in regulating motivational drives
* discusses physiological needs, particularly thirst, hunger, and sexual motivation

Take-home messages:
* The brain is as much about motivation and emotion as it is about cognition and thinking
* Biological urges are underestimated motivational forces when we are not currently experiencing them

==Outline==
[[File:DALL-E Generative AI Brain VanGogh.png|thumb|It is easy to overlook the brain's involvement in [[motivation and emotion]] because it is hidden beneath the skulls, skin, hair, and often adornments. But what if our brain activity became more observable, how would that affect understanding of our motivation and emotion?]]

[[File:Hunger strike - Day 53.JPG|thumb|right|290px|Physiological needs such as breathing, drinking, urinating, eating, defecating, and sleeping are often overlooked as motivational forces until they range outside of [[w:Homeostasis|homeostasis]] and then become increasingly urgemt amd motivationally demanding. It takes extreme motivation, for example, to go on an extended hunger strike.]]
;Motivated and emotional brain
* Neuroscience
* Brain structures
* Subcortical
** Reticular formation
** Amygdala
**Reward centre
**Basal ganglia
**Hypothalamus
* Cortical
** Insula
** Prefrontal cortex
** Orbitofrontal cortex
** Ventromedial PFC
** Dorsolateral PFC
** Anterior cingulate cortex
* Bidirectional
** Neurotransmitters
** Dopamine
** Serotonin
** Norepinephrine
** Endorphins
*Hormones
** Cortisol
** Oxytocin
** Testosterone
** Ghrelin (Part B)
** Leptin (Part B)

;Physiological needs
* Needs
* Regulatory processes
* Example physiological needs
** Thirst
** Hunger
** Sexual motivation

==Focus==
This lecture highlights specific brain structures and communication pathways that psychological science has identified as contributing to the subjective experience of various motivational and emotional states.

==3D brain model==
* Learn about the location and function of key brain structures using [https://www.brainfacts.org/3d-brain 3d brain] (brainfacts.org)
* This 3D, interactive model of the human brain shows the main structures and explains their functions.
* Task: Can you find each of the brain structures mentioned in this lecture in the 3D model?

==Readings==
* Chapter 03: The motivated and emotional brain ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]] or [[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]])
* Chapter 04: Physiological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 4: Biological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2024]])

==Slides==
&lt;!-- ** [https://docs.google.com/presentation/d/1wNaegpzIkQ4XyeRcN9BRXQ1gGNR5XX3cG7x_dtBGj6c/edit?usp=sharing Lecture 01 and 02 recap] (Google Slides) --&gt;
* [https://docs.google.com/presentation/d/1oI8g-0xvSxETUwYOW1TLsRJdiSq3AbVq6YMlm8D3ivc/edit?usp=sharing Motivated and emotional brain] (Google Slides)
* [https://docs.google.com/presentation/d/1LgYQ9ydIaj5AJZEW7MkH1M2zVKxjWQe4vetZnOairQE/edit?usp=sharing Physiological needs] (Google Slides)
&lt;!-- 
** [https://www.slideshare.net/jtneill/motivation-and-emotion-introduction-and-historical-perspectives-recap Lecture 01 and 02 recap] (Slideshare)
** [https://www.slideshare.net/jtneill/motivated-and-emotional-brain Motivated and emotional brain] (Slideshare)
** [https://www.slideshare.net/jtneill/physiological-needs Physiological needs] (Slideshare) --&gt;
&lt;!-- * [http://www.slideshare.net/jtneill/brain-and-physiological-needs Lecture slides] (Slideshare)
* Handouts
** [[Media:Brain and physiological needs 6 slides per page.pdf|Download 6 slides to a page]]: [[File:Brain and physiological needs 6 slides per page.pdf|100px]]
** [[Media:Brain and physiological needs 3 slides per page.pdf|Download 3 slides to a page]]:[[File:Brain and physiological needs 3 slides per page.pdf|100px]]
--&gt;

==See also==
;Wikiversity
* [[/Images/]]
* [[Motivation and emotion/Brain structures|Brain structures]]
* [[Motivation and emotion/Hormones|Hormones]]
* [[Motivation and emotion/Neurotransmitters|Neurotransmitters]]
* Book chapters
** [[:Category:Motivation and emotion/Book/Brain|Brain]] (Category)
** [[:Category:Motivation and emotion/Book/Hormones|Hormones]] (Category)
** [[:Category:Motivation and emotion/Book/Neurotransmitters|Neurotransmitters]] (Category)
** [[:Category:Motivation and emotion/Book/Needs/Physiological|Physiological needs]] (Category)&lt;!--
[[Motivation and emotion/Book/2025/Thirst regulation|Thirst regulation]] --&gt;

;Wikipedia
* [[w:Autonomic nervous system|Autonomic nervous system]]
* [[w:ERG theory|ERG theory]]
* [[w:Limbic system|Limbic system]]
* [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]]
* [[w:Nucleus (neuroanatomy)|Nucleus (neuroanatomy)]]
* [[w:Parasympathetic nervous system|Parasympathetic nervous system]]
* [[w:Prefrontal cortex|Prefrontal cortex]]
* [[w:Reward system|Reward system]]
* [[w:Sympathetic nervous system|Sympathetic nervous system]]

;Lectures
* [[{{#titleparts:{{PAGENAME}}|2}}/Historical development and assessment skills|Historical development and assessment skills]] (Previous lecture)
* [[{{#titleparts:{{PAGENAME}}|2}}/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] (Next lecture)

;Tutorials
* [[Motivation and emotion/Tutorials/Physiological needs|Physiological needs]]
&lt;!--
==References==
{{Hanging indent|1=
Australian Bureau of Statistics (2013). [http://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/4338.0~2011-13~Main%20Features~Overweight%20and%20obesity~10007 Overweight and obesity]. ''4338.0 - Profiles of Health, Australia, 2011-13''.
Eder, A. B., Elliot, A. J., &amp; Harmon-Jones, E. (2013). [http://emr.sagepub.com/content/5/3/227  Approach and avoidance motivation: Issues and advances]. ''Emotion Review'', ''5''(3), 308-311. https://doi.org/10.1177/1754073913477990.}}
--&gt;

==Recording==
* [https://au-lti.bbcollab.com/recording/54f3cdb5b30a476fbcbb77824a1b9dfb Lecture 03] (2025)&lt;!--
* [https://au-lti.bbcollab.com/recording/b8834e9830314aa3b804d3c6c3e7a740 Lecture 03] (2024)
* [https://au-lti.bbcollab.com/recording/546476bf547f4efd8ae55b05e4547efc Lecture 03] (2023)
* [https://au-lti.bbcollab.com/recording/17f200f050e044da9a6571ffdf63c78c Lecture 03] (2022)
* [https://au-lti.bbcollab.com/recording/d34da988d75c48b99df662329594cc9f Lecture 03] (2021)
--&gt;

==References==
{{Hanging indent|1=

Saper, C. B., &amp; Lowell, B. B. (2014). The hypothalamus. ''Current Biology'', ''24''(23), R1111–R1116. https://doi.org/10.1016/j.cub.2014.10.023

}}

==External links==
* [https://fs.blog/knowledge-project-podcast/anna-lembke/ Between pleasure and pain] (Dr. Anna Lembke, The Knowledge Project Ep. #159)
* [https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/short-stuff-hangry-102038598/ Hangry] (Stuff You Should Know, Podcast, 12:30 mins)
* [https://www.youtube.com/watch?v=tZ4YnYUJnOQ&amp;list=PL9JAHwJN4qyArhEyLUgU_MoGddk2PVTeb Hormones of hunger: Leptin and ghrelin] (Corporis, 2019, YouTube, 9:33 mins) - how leptin and ghrelin work together to modulate hunger&lt;!-- As you watch the video, consider: What causes hunger and eating? --&gt;
* [https://www.ted.com/playlists/1/how_does_my_brain_work How does my brain work?] (TED Talks playlist)
* [https://www.youtube.com/watch?v=Qymp_VaFo9M Let's talk about sex] (Crash Course Psychology #27; YouTube 11:35 mins)
* [https://www.ted.com/talks/david_anderson_your_brain_is_more_than_a_bag_of_chemicals Your brain is more than a bag of chemicals] (David Anderson, 2013, TED talk, 16 mins) - neuroscientific research into motivation and emotion using a basic animal model (fruit fly)&lt;!-- As you watch the video, some questions to think about:

1. Do animals experience emotions? If so, which emotions - and why?
2. What might pharmacological treatment of psychological disorders look like in 20, 50, 100 years? --&gt;
{{Motivation and emotion/Lectures/Navigation}}

[[Category:Motivation and emotion/Lectures/Brain and physiological needs]]</text>
      <sha1>odhd4utdlvz9j1p4x8o1s5q0oyyab90</sha1>
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    <revision>
      <id>2810727</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
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      <text bytes="8579" sha1="7ar36bakc4gmaospg775d6fxe2bpgrq" xml:space="preserve">{{Motivation and emotion/Lectures|Lecture 03: Brain and physiological needs|third}}
{{Motivation and emotion/Lectures/Complete}}
&lt;!-- {{Motivation and emotion/Lectures/In development}} --&gt;
&lt;!-- {{Motivation and emotion/Lectures/Complete}} --&gt;
[[File:WP20Symbols brain.svg|right|250px]]
==Overview==
This lecture:
* explains the role of [[Motivation and emotion/Brain structures|brain structures]], [[Motivation and emotion/Neurotransmitters|neurotransmitters]], and [[Motivation and emotion/Hormones|hormones]] in regulating motivational drives
* discusses physiological needs, particularly thirst, hunger, and sexual motivation

Take-home messages:
* The brain is as much about motivation and emotion as it is about cognition and thinking
* Biological urges are underestimated motivational forces when we are not currently experiencing them

==Outline==
[[File:DALL-E Generative AI Brain VanGogh.png|thumb|right|250px|It is easy to overlook the brain's involvement in [[motivation and emotion]] because it is hidden beneath the skull, skin, hair, and adornments. But what if our brain activity became more observable, how would that affect understanding of our motivation and emotion?]]

[[File:Hunger strike - Day 53.JPG|thumb|right|250px|Physiological needs such as breathing, drinking, urinating, eating, defecating, and sleeping are often overlooked as motivational forces until they range outside of [[w:Homeostasis|homeostasis]] and then become increasingly urgemt amd motivationally demanding. It takes extreme motivation, for example, to go on an extended hunger strike.]]
;Motivated and emotional brain
* Neuroscience
* Brain structures
* Subcortical
** Reticular formation
** Amygdala
**Reward centre
**Basal ganglia
**Hypothalamus
* Cortical
** Insula
** Prefrontal cortex
** Orbitofrontal cortex
** Ventromedial PFC
** Dorsolateral PFC
** Anterior cingulate cortex
* Bidirectional
** Neurotransmitters
** Dopamine
** Serotonin
** Norepinephrine
** Endorphins
*Hormones
** Cortisol
** Oxytocin
** Testosterone
** Ghrelin (Part B)
** Leptin (Part B)

;Physiological needs
* Needs
* Regulatory processes
* Example physiological needs
** Thirst
** Hunger
** Sexual motivation

==Focus==
This lecture highlights specific brain structures and communication pathways that psychological science has identified as contributing to the subjective experience of various motivational and emotional states.

==3D brain model==
* Learn about the location and function of key brain structures using [https://www.brainfacts.org/3d-brain 3d brain] (brainfacts.org)
* This 3D, interactive model of the human brain shows the main structures and explains their functions.
* Task: Can you find each of the brain structures mentioned in this lecture in the 3D model?

==Readings==
* Chapter 03: The motivated and emotional brain ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]] or [[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]])
* Chapter 04: Physiological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 4: Biological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2024]])

==Slides==
&lt;!-- ** [https://docs.google.com/presentation/d/1wNaegpzIkQ4XyeRcN9BRXQ1gGNR5XX3cG7x_dtBGj6c/edit?usp=sharing Lecture 01 and 02 recap] (Google Slides) --&gt;
* [https://docs.google.com/presentation/d/1oI8g-0xvSxETUwYOW1TLsRJdiSq3AbVq6YMlm8D3ivc/edit?usp=sharing Motivated and emotional brain] (Google Slides)
* [https://docs.google.com/presentation/d/1LgYQ9ydIaj5AJZEW7MkH1M2zVKxjWQe4vetZnOairQE/edit?usp=sharing Physiological needs] (Google Slides)
&lt;!-- 
** [https://www.slideshare.net/jtneill/motivation-and-emotion-introduction-and-historical-perspectives-recap Lecture 01 and 02 recap] (Slideshare)
** [https://www.slideshare.net/jtneill/motivated-and-emotional-brain Motivated and emotional brain] (Slideshare)
** [https://www.slideshare.net/jtneill/physiological-needs Physiological needs] (Slideshare) --&gt;
&lt;!-- * [http://www.slideshare.net/jtneill/brain-and-physiological-needs Lecture slides] (Slideshare)
* Handouts
** [[Media:Brain and physiological needs 6 slides per page.pdf|Download 6 slides to a page]]: [[File:Brain and physiological needs 6 slides per page.pdf|100px]]
** [[Media:Brain and physiological needs 3 slides per page.pdf|Download 3 slides to a page]]:[[File:Brain and physiological needs 3 slides per page.pdf|100px]]
--&gt;

==See also==
;Wikiversity
* [[/Images/]]
* [[Motivation and emotion/Brain structures|Brain structures]]
* [[Motivation and emotion/Hormones|Hormones]]
* [[Motivation and emotion/Neurotransmitters|Neurotransmitters]]
* Book chapters
** [[:Category:Motivation and emotion/Book/Brain|Brain]] (Category)
** [[:Category:Motivation and emotion/Book/Hormones|Hormones]] (Category)
** [[:Category:Motivation and emotion/Book/Neurotransmitters|Neurotransmitters]] (Category)
** [[:Category:Motivation and emotion/Book/Needs/Physiological|Physiological needs]] (Category)&lt;!--
[[Motivation and emotion/Book/2025/Thirst regulation|Thirst regulation]] --&gt;

;Wikipedia
* [[w:Autonomic nervous system|Autonomic nervous system]]
* [[w:ERG theory|ERG theory]]
* [[w:Limbic system|Limbic system]]
* [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]]
* [[w:Nucleus (neuroanatomy)|Nucleus (neuroanatomy)]]
* [[w:Parasympathetic nervous system|Parasympathetic nervous system]]
* [[w:Prefrontal cortex|Prefrontal cortex]]
* [[w:Reward system|Reward system]]
* [[w:Sympathetic nervous system|Sympathetic nervous system]]

;Lectures
* [[{{#titleparts:{{PAGENAME}}|2}}/Historical development and assessment skills|Historical development and assessment skills]] (Previous lecture)
* [[{{#titleparts:{{PAGENAME}}|2}}/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] (Next lecture)

;Tutorials
* [[Motivation and emotion/Tutorials/Physiological needs|Physiological needs]]
&lt;!--
==References==
{{Hanging indent|1=
Australian Bureau of Statistics (2013). [http://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/4338.0~2011-13~Main%20Features~Overweight%20and%20obesity~10007 Overweight and obesity]. ''4338.0 - Profiles of Health, Australia, 2011-13''.
Eder, A. B., Elliot, A. J., &amp; Harmon-Jones, E. (2013). [http://emr.sagepub.com/content/5/3/227  Approach and avoidance motivation: Issues and advances]. ''Emotion Review'', ''5''(3), 308-311. https://doi.org/10.1177/1754073913477990.}}
--&gt;

==Recording==
* [https://au-lti.bbcollab.com/recording/54f3cdb5b30a476fbcbb77824a1b9dfb Lecture 03] (2025)&lt;!--
* [https://au-lti.bbcollab.com/recording/b8834e9830314aa3b804d3c6c3e7a740 Lecture 03] (2024)
* [https://au-lti.bbcollab.com/recording/546476bf547f4efd8ae55b05e4547efc Lecture 03] (2023)
* [https://au-lti.bbcollab.com/recording/17f200f050e044da9a6571ffdf63c78c Lecture 03] (2022)
* [https://au-lti.bbcollab.com/recording/d34da988d75c48b99df662329594cc9f Lecture 03] (2021)
--&gt;

==References==
{{Hanging indent|1=

Saper, C. B., &amp; Lowell, B. B. (2014). The hypothalamus. ''Current Biology'', ''24''(23), R1111–R1116. https://doi.org/10.1016/j.cub.2014.10.023

}}

==External links==
* [https://fs.blog/knowledge-project-podcast/anna-lembke/ Between pleasure and pain] (Dr. Anna Lembke, The Knowledge Project Ep. #159)
* [https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/short-stuff-hangry-102038598/ Hangry] (Stuff You Should Know, Podcast, 12:30 mins)
* [https://www.youtube.com/watch?v=tZ4YnYUJnOQ&amp;list=PL9JAHwJN4qyArhEyLUgU_MoGddk2PVTeb Hormones of hunger: Leptin and ghrelin] (Corporis, 2019, YouTube, 9:33 mins) - how leptin and ghrelin work together to modulate hunger&lt;!-- As you watch the video, consider: What causes hunger and eating? --&gt;
* [https://www.ted.com/playlists/1/how_does_my_brain_work How does my brain work?] (TED Talks playlist)
* [https://www.youtube.com/watch?v=Qymp_VaFo9M Let's talk about sex] (Crash Course Psychology #27; YouTube 11:35 mins)
* [https://www.ted.com/talks/david_anderson_your_brain_is_more_than_a_bag_of_chemicals Your brain is more than a bag of chemicals] (David Anderson, 2013, TED talk, 16 mins) - neuroscientific research into motivation and emotion using a basic animal model (fruit fly)&lt;!-- As you watch the video, some questions to think about:

1. Do animals experience emotions? If so, which emotions - and why?
2. What might pharmacological treatment of psychological disorders look like in 20, 50, 100 years? --&gt;
{{Motivation and emotion/Lectures/Navigation}}

[[Category:Motivation and emotion/Lectures/Brain and physiological needs]]</text>
      <sha1>7ar36bakc4gmaospg775d6fxe2bpgrq</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Numerical Methods</title>
    <ns>0</ns>
    <id>119778</id>
    <revision>
      <id>2810626</id>
      <parentid>2809805</parentid>
      <timestamp>2026-05-20T17:53:08Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2810626</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="ncwo3aqs7he1jlfb1zcz25pvf15rklt" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260513.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

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

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

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

== Matrix Algebra == 

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


&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

== Regression ==

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

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

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


== Interpolation ==

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

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

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


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

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

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


&lt;/br&gt;


== Using Symbolic Package in Octave ==

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

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

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

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

== Read some tutorials about symbolic computation ==

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

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

[[Category:Numerical methods]]

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


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>ncwo3aqs7he1jlfb1zcz25pvf15rklt</sha1>
    </revision>
    <revision>
      <id>2810628</id>
      <parentid>2810626</parentid>
      <timestamp>2026-05-20T17:54:43Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2810628</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="h517c8ti8xkq7y7v56bevbif9hbtw1d" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260519.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

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

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

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

== Matrix Algebra == 

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


&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

== Regression ==

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

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

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


== Interpolation ==

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

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

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


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

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

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


&lt;/br&gt;


== Using Symbolic Package in Octave ==

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

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

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

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

== Read some tutorials about symbolic computation ==

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

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

[[Category:Numerical methods]]

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


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>h517c8ti8xkq7y7v56bevbif9hbtw1d</sha1>
    </revision>
    <revision>
      <id>2810630</id>
      <parentid>2810628</parentid>
      <timestamp>2026-05-20T17:55:35Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2810630</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="15lcvw13q0h2990wgho0xd9im0afcj7" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260520.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

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

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

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

== Matrix Algebra == 

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


&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

== Regression ==

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

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

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


== Interpolation ==

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

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

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


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

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

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


&lt;/br&gt;


== Using Symbolic Package in Octave ==

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

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

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

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

== Read some tutorials about symbolic computation ==

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

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

[[Category:Numerical methods]]

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


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>15lcvw13q0h2990wgho0xd9im0afcj7</sha1>
    </revision>
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  <page>
    <title>Wind</title>
    <ns>0</ns>
    <id>121519</id>
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      <id>2810615</id>
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      <timestamp>2026-05-20T15:04:04Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>project box(es)</comment>
      <origin>2810615</origin>
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      <text bytes="2390" sha1="hoimqwl0falxl85xnjnjuja5tnt8zlu" xml:space="preserve">{{environmental science}}
[[File:Santa Ana winds - satellite image.jpg|thumb|The wind from the desert pushing dust and smoke far out into the Pacific Ocean.]]On [[Earth]], '''wind''' is mostly the movement of air. In outer space, solar wind is the movement of gases or particles from the [[sun]] through space. The strongest winds seen on a planet in our solar system are on Neptune and Saturn.

Short bursts of fast winds are called ''gusts''. Strong winds that go on for about one minute are called ''squalls''. Winds that go on for a long time are called many different things, such as ''breeze'', ''gale'', ''[[hurricane]]'', and ''typhoon''.

Wind can move land, especially in deserts. Cold wind can sometimes have a bad effect on livestock. Wind also affects animals' food stores, their hunting and the way they protect themselves.

High winds can cause damage depending on how strong they are. Sometimes gusts of wind can make poorly made [[bridge structure|bridges]] move or be destroyed, like the Tacoma Narrows Bridge in 1940.&lt;ref&gt;{{cite book
|author=T. P. Grazulis
|year=2001
|url=http://books.google.com/?id=N6Tiz_7VmJoC&amp;pg=PA127&amp;lpg=PA127&amp;dq=wind+sensitivity+of+powerlines
|title=The tornado
|publisher=University of Oklahoma Press
|pages=126–127
|isbn=9780806132587
|accessdate=2009-05-13
}}&lt;/ref&gt; Power can go out because of wind, even if its speed is as low. This is because tree branches could change the flow of energy through power lines.&lt;ref&gt;{{cite book
|author=Hans Dieter Betz
|author2=Ulrich Schumann
|author3=Pierre Laroche
|year=2009
|url=http://books.google.com/?id=U6lCL0CIolYC&amp;pg=PA187&amp;lpg=PA187&amp;dq=Spatial+Distribution+and+Frequency+of+Thunderstorms+and+Lightning+in+Australia+wind+gust|title=Lightning: Principles, Instruments and Applications
|publisher=Springer
|pages=202–203
|isbn=9781402090783
|accessdate=2009-05-13
}}&lt;/ref&gt; No species of tree can resist hurricane-force winds, but trees with roots that are not very deep can be blown over more easily. Trees such as eucalyptus, sea hibiscus, and avocado are brittle and are damaged more easily.&lt;ref&gt;{{cite web
|author=Derek Burch
|url=http://edis.ifas.ufl.edu/EP042
|title=How to Minimize Wind Damage in the South Florida Garden
|publisher=University of Florida
|date=2006-04-26
|accessdate=2009-05-13
}}&lt;/ref&gt;
{{commons}}
== References ==
{{Reflist}}
{{wind}}
[[Category:Wind]]
[[Category:Weather]]</text>
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  <page>
    <title>User talk:Egm4313.s12</title>
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      <comment>/* You may be eligible to vote in the U4C election */ new section</comment>
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      <text bytes="11850" sha1="1rfb7xbg32wnh3ex8mv6rwgdk89dqst" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello Egm4313.s12, and [[Wikiversity:Welcome|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[File:Insert-signature.png]] in the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy policy]], [[Wikiversity:Civility|Civility policy]], and the [[Foundation:Terms of Use|Terms of Use]] while at Wikiversity.

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You don't need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:HappyCamper|HappyCamper]] 20:10, 15 December 2011 (UTC)&lt;/div&gt;
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== Reply from a custodian... ==
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[http://en.wikiversity.org/wiki/Wikiversity:Request_custodian_action Request custodian action]
--&gt;
[[Wikiversity:Request_custodian_action | Request custodian action]]
:Hello, I [http://en.wikiversity.org/w/index.php?title=Wikiversity%3ARequest_custodian_action&amp;action=historysubmit&amp;diff=845695&amp;oldid=840240 got your note] about needing a custodian to certify your students' accounts, but I don't understand what you mean by this. I'm sure it's something straightforward that other custodians have done before. Please let me know how I can help. --[[User:HappyCamper|HappyCamper]] 20:10, 15 December 2011 (UTC)

Hi, thanks for certifying this account.  this kind of certification is what i meant.  once a student opens an account, s/he has to wait till a custodian like you certify it so that s/he can begin to upload files and edit articles without having to keep typing in the captcha expression.  thanks for a quick certification of this new account.
[[User:Egm4313.s12|Egm4313.s12]] 22:30, 15 December 2011 (UTC)

:No problem. I posted a followup [http://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;diff=prev&amp;oldid=845737 here]. I'm pretty sure there is a grace period of 4 days and 10 edits before a new account can upload files to Wikiversity, so this is something to take into account. The [[Wikiversity:Sandbox|sandbox]] might be a good place for making some of these edits. Also, if you run into other issues on Wikiversity, please do let us know, we're glad to assist. --[[User:HappyCamper|HappyCamper]] 05:04, 16 December 2011 (UTC)

ok, great.  now i understand the process.  before i associated the greeting message with the ability to upload after 4 days.  but your explanation clarifies the process.  i now have a better explanation to students.  i also cross-post [http://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;oldid=845781 here].
[[User:Egm4313.s12|Egm4313.s12]] 11:36, 16 December 2011 (UTC)

== Image uploads ==

Hi Professor. You may recall I've previously tried to clear up some confusion as to the licensing status of your uploads. With reference to your two recent images ([[:File:Iea.s12.damping.svg]] and [[:File:Pea2.s12.team.structure.svg]]), I note that you have used the {{tl|CC-by-sa-3.0-dual}} licence tag. I've proposed this is deleted because it creates a confusing situation. 

Can you clarify whether you are releasing these files under the Creative Commons Attribution ShareAlike 3.0 '''OR''' the Creative Commons Attribution NonCommercial ShareAlike 3.0 licence. I doesn't really seem possible that you can choose both since one permits commercial reuse and the other does not so anyone wanting to reuse them commercial could ignore the more restrictive licence.

It is also unclear as to why you refer to &quot;see also [[w:Template:MultiLicenseWithCC-BySA-Any|MultiLicenseWithCC-BySA-Any]]&quot; since that template deals with text contributions rather than images.  

If this situation cannot be resolved then these and the many other images uploaded by you and your students with similar circumstances may have to be deleted since the non-commercial restriction confusion is a problem. [[User:Adambro|Adambro]] 16:26, 2 January 2012 (UTC)

: i prefer the Creative Commons Attribution NonCommercial ShareAlike 3.0 licence for all images.  please change accordingly.  do inform the students before taking any action.  thanks. ps: this template was put in my uploads by a bot; it was not my original licensing template; then after i just use the template that the bot put in. [[User:Egm6322.s12|Egm6322.s12]] 23:00, 10 January 2012 (UTC)
::Yes, I'm aware a bot did replace what you entered originally but I think the problem was there to begin with and still exists on your recent uploads such as [[:File:Pea1.f11.mtg42.djvu]]. You've said that you prefer the Creative Commons Attribution NonCommercial ShareAlike 3.0 licence for all images but what exactly does that mean? Are you not prepared to allow commercial resuse, in which case you should only mention Attribution Share-Alike version 3.0 but in that situation the images would likely need to be deleted. If you are prepared to allow commercial reuse, then perhaps you would be better not mentioning Creative Commons Attribution NonCommercial ShareAlike 3.0. That would avoid the confusion created by the current situation where it isn't clear whether commercial use is actually permitted. [[User:Adambro|Adambro]] 21:11, 12 January 2012 (UTC)

images and text are alike, and should be treated with the same dual license.  the dual license allows one to use one license or the other license.  if someone wants to make money on our work, they can select the license without NC.  [[User:Egm6322.s12|Egm6322.s12]] 15:16, 13 January 2012 (UTC)

== Licensing and templates ==

Hi, I did get your message that you left on my talk page and have spent the week trying to get more information on the matter. I still need more time before I get back to you on the licensing / template issue above. --[[User:HappyCamper|HappyCamper]] 16:08, 19 January 2012 (UTC)

==Student work should be in user space, not main space - please move==
Hi Loc,
Could you please move your students' work from the main space into their user spaces? For example, most of the content found under a search for &quot;Egm4313.s12.team&quot; is currently located in the main space: http://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;profile=all&amp;search=Egm4313.s12.team&amp;fulltext=Search Main space is for learning resources. Individual student work should be in the user space unless it has broader educational value. I have moved some content (e.g., http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=move&amp;user=Jtneill&amp;page=&amp;year=&amp;month=-1&amp;tagfilter=), but there is much more. Could you please follow up and move the rest of your students' content? Let me know if you'd like to discuss. Sincerely, James. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:33, 11 April 2012 (UTC)
== Files Missing Information ==
Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information.
{{colbegin|3}}
* [[:File:MoM S13 lecture summaries, notes, audios, videos.pdf]]
* [[:File:Mom.s13.sec0.pdf]]
* [[:File:Mom.s13.sec1.pdf]]
* [[:File:Mom.s13.sec10.pdf]]
* [[:File:Mom.s13.sec11 - Copy.pdf]]
* [[:File:Mom.s13.sec11.pdf]]
* [[:File:Mom.s13.sec12.pdf]]
* [[:File:Mom.s13.sec13 - Copy.pdf]]
* [[:File:Mom.s13.sec13.pdf]]
* [[:File:Mom.s13.sec14 - Copy.pdf]]
* [[:File:Mom.s13.sec14.pdf]]
* [[:File:Mom.s13.sec15 - Copy.pdf]]
* [[:File:Mom.s13.sec15.pdf]]
* [[:File:Mom.s13.sec2 - Copy.pdf]]
* [[:File:Mom.s13.sec2.pdf]]
* [[:File:Mom.s13.sec3.pdf]]
* [[:File:Mom.s13.sec4 - Copy.pdf]]
* [[:File:Mom.s13.sec4.pdf]]
* [[:File:Mom.s13.sec5 - Copy.pdf]]
* [[:File:Mom.s13.sec5.pdf]]
* [[:File:Mom.s13.sec51.pdf]]
* [[:File:Mom.s13.sec6 - Copy.pdf]]
* [[:File:Mom.s13.sec6.pdf]]
* [[:File:Mom.s13.sec7.pdf]]
* [[:File:Mom.s13.sec8.pdf]]
* [[:File:Mom.s13.sec9.pdf]]
{{colend}}
[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 04:53, 19 February 2017 (UTC)
== Files Missing Information ==
Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information.
{{colbegin|3}}
* [[:File:Chrome 2020-02-09 19-46-56.png]]
{{colend}}
[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 01:06, 11 February 2020 (UTC)

== You may be eligible to vote in the U4C election ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
I am contacting you because you previously voted in elections related to the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee (U4C)]]. You may be eligible to vote in the current U4C election, which is open now and closes on 2 June 2026. You can find out more about the candidates and the election on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|the election page on Meta]], and from there you can access the vote itself. Your participation in these elections is important to the governance of Wikimedia communities, and your time spent learning about the candidates and voting is appreciated.

-- In cooperation with the U4C, [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]])&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:Keegan (WMF)|Keegan (WMF)]] ([[m:User_talk:Keegan (WMF)|talk]]) 17:16, 20 May 2026 (UTC)
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ some clarity about distinction between deletion nomination options</comment>
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      <text bytes="1582" sha1="1z5ptn3mg7fo6gwblrt364sz7ndw8pz" xml:space="preserve">{{Documentation subpage}}
&lt;!-- PLEASE ADD CATEGORIES AND INTERWIKIS AT THE BOTTOM OF THIS PAGE. --&gt;
{{tsh|rfd}}
== Usage ==
Place the following at the top of resources to inform participants of a discussion at [[Wikiversity:Requests for Deletion]]: 

 {{tlx|Deletion request}}

Resources are sorted into [[:Category:Requests for Deletion]].

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[[fr:Modèle:PàS]]
[[sv:Mall:Sffr]]
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  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
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      <timestamp>2026-05-20T13:51:37Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2810591</origin>
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      <text bytes="3135" sha1="88fqoxyb9nvnsrv58p1isv8e3hju38n" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260520.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260520.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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  <page>
    <title>Film and Documentary Studies</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <minor />
      <comment>cat(s)</comment>
      <origin>2810614</origin>
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      <text bytes="2474" sha1="qvwoex9kmclkaped5gvmde7i3qcf2y8" xml:space="preserve">== Documentary Studies ==
{|class=&quot;wikitable sortable&quot; border=&quot;2&quot;
!Lfd. !! Titel !!  Abstract !! Bewertung
|-
| 1
|| [https://www.youtube.com/watch?v=pOfUoKELb_U&amp;feature=plcp COM 351 Lecture 01 - Getting Started  ]
|| &lt;small&gt;This course will allow students to study the methods by which documentary work is conducted and to complete a documentary project of their own. The course will connect the qualitative methods of the social sciences and the humanistic concerns of the arts by allowing students to study documentary subjects as captured by non-fiction, photography, film, tape recorder, and the World Wide Web. Special emphasis will be placed on narrative and metaphor.&lt;/small&gt;
|| 966 views
|-
| 2
|| [https://www.youtube.com/watch?v=htJbTTrSCuU&amp;feature=bf_prev&amp;list=PL89535D5C116F41CC COM 351 Lecture 02 - Connections for Research in the Social Sciences and the Humanities]
|| &lt;small&gt;Connections to social sciences&lt;/small&gt;
|| 1240 views
|-
| 3
|| [https://www.youtube.com/watch?v=sidZzYkJbpo&amp;feature=relmfu COM 351 Lecture 03 - A Guide to Philosophical Decision-Making]
|| &lt;small&gt;Ethical Codes in documentary.&lt;/small&gt;
|| 397 views
|-
| 4
|| [https://www.youtube.com/watch?v=AesX4lDdtYM&amp;feature=relmfu COM 351 Lecture 04 - The Research Proposal for a Documentary Project]
|| &lt;small&gt;Key concepts in research. How to operationalize research concepts ?&lt;/small&gt;
|| 2101 views
|-
| 5
|| [https://www.youtube.com/watch?v=rUYeb_1cG2M&amp;feature=relmfu COM 351 Lecture 05 - Model Construction and the Preparation of Field Research]
|| &lt;small&gt;Design of your research&lt;/small&gt;
|| 459 views
|-
| 6
|| [https://www.youtube.com/watch?v=jtNh1bi4jcg&amp;feature=relmfu COM 351 Lecture 06 - Masters and Interpretation]
|| &lt;small&gt;Writing in documentary studies&lt;/small&gt;
|| 161 views
|-
| 7
|| [https://www.youtube.com/watch?v=uioHMWwfOf4&amp;feature=relmfu COM 351 Lecture 07 - Photography and Documentary Studies]
|| &lt;small&gt;&lt;/small&gt;
|| 2079 views
|-
| 8
|| [https://www.youtube.com/watch?v=4iITUomPhto&amp;feature=relmfu COM 351 Lecture 08 - Oral History and Documentary Studies]
|| &lt;small&gt;Oral history&lt;/small&gt;
|| 402 views
|-
| 9
|| [https://www.youtube.com/watch?v=IJwosULqmoc&amp;feature=relmfu COM 351 Lecture 09 - Film and Documentary Studies]
|| &lt;small&gt;Film documentary&lt;/small&gt;
|| 1899 views
|-
| 10
||[https://www.youtube.com/watch?v=CPxql4WM_uA&amp;feature=relmfuCOM 351 Lecture 10 - The World Wide Web and Documentary Studies]
|| &lt;small&gt;The world wide web&lt;/small&gt;
|| 118 views
|-
|}
[[Category:Film]]</text>
      <sha1>qvwoex9kmclkaped5gvmde7i3qcf2y8</sha1>
    </revision>
  </page>
  <page>
    <title>IRC</title>
    <ns>0</ns>
    <id>156122</id>
    <revision>
      <id>2810612</id>
      <parentid>2786725</parentid>
      <timestamp>2026-05-20T14:48:06Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>more appropriate for WV</comment>
      <origin>2810612</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="238" sha1="bo8y2hcfa27q25xdjl23fj9onh55dt4" xml:space="preserve">Internet Relay Chat (IRC) is a system that facilitates transfer of messages in the form of text.&lt;ref&gt;[[Wikipedia: Internet Relay Chat]]&lt;/ref&gt;

==Resources==

* [[IRC as an online community]]

== References ==
{{Reflist}}

[[Category:IRC]]</text>
      <sha1>bo8y2hcfa27q25xdjl23fj9onh55dt4</sha1>
    </revision>
  </page>
  <page>
    <title>Nanoengineering</title>
    <ns>0</ns>
    <id>161280</id>
    <revision>
      <id>2810621</id>
      <parentid>2234355</parentid>
      <timestamp>2026-05-20T16:56:52Z</timestamp>
      <contributor>
        <username>~2026-30360-41</username>
        <id>3079153</id>
      </contributor>
      <comment>/* Discussion questions */</comment>
      <origin>2810621</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="766" sha1="abg8yqi1zdp20h868wqlglrb6jlhvsk" xml:space="preserve">Nanoengineering is the practice of engineering on the nanoscale. or one billionth of a meter.  Nanoengineering is largely a synonym for nanotechnology, but emphasizes the engineering rather than the pure science aspects of the field.&lt;ref&gt;[[Wikipedia: Nanoengineering]]&lt;/ref&gt;
__NOTOC__
==Discussion questions==
* How can nano-engineering be used to benefit all of humanity? 
* What sort of problems can nano-engineering potentially solve?
* Can nano-engineering be used to eliminate diseases such as COVID-19 and Cancer?

==Readings==
===Wikipedia===
* [[w:Nanomedicine|Nanomedicine]]
* [[w:Green nanotechnology|Green nanotechnology]]
* [[w:Nanoengineering|Nanoengineering]]
* [[w:Nanorobotics|Nanorobotics]]

== References ==
{{Reflist}}

[[Category:Nanotechnology]]</text>
      <sha1>abg8yqi1zdp20h868wqlglrb6jlhvsk</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2810596</id>
      <parentid>2810443</parentid>
      <timestamp>2026-05-20T14:06:05Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2810596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="g3kat9f02vd3xu9su6sthdgz815sky4" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260520.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
      <sha1>g3kat9f02vd3xu9su6sthdgz815sky4</sha1>
    </revision>
  </page>
  <page>
    <title>User:C.Koltzenburg/Wikipedia entries on fiction and non-propositional knowledge representation</title>
    <ns>2</ns>
    <id>200619</id>
    <revision>
      <id>2810640</id>
      <parentid>1421491</parentid>
      <timestamp>2026-05-20T18:30:12Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>Atcovi moved page [[Wikipedia entries on fiction and non-propositional knowledge representation]] to [[User:C.Koltzenburg/Wikipedia entries on fiction and non-propositional knowledge representation]] without leaving a redirect: moving under userspace; underdeveloped and hasn't been touched in over a decade</comment>
      <origin>1421491</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2991" sha1="19ht12x6bsby9ltpsxwmx300wadc96g" xml:space="preserve">''Wikipedia entries on fiction: Under what terms is it possible to report about non-propositional knowledge gained from reading?'' (Abstract) (For the full text of the PhD thesis see ''[[:de:Nicht-propositionales Wissen aus Literaturlektüre und Bedingungen seiner Darstellbarkeit in Wikipedia-Einträgen zu literarischen Werken]]'')

(Summary: Given that non-propositional knowledge gained from reading a work of art is fiction's defining characteristic, on what terms can entries on fiction be improved?)

Given that Wikipedia entries are likely to extert a strong influence on how literary texts are perceived – due to their preferential ranking in Google – there is some demand that research dealing with the transfer of knowledge on literature to the public be more concerned with looking into both the content that is available for free on the Web and any aspect that may come with writing about literature for free. This contribution argues from within Wikipedia's multidisciplinary consensus-driven space in which propositional knowledge is given priority that it would be essential for entries on fiction to present non-propositional knowledge as being one of its hallmarks. For this aim, a special concept is developed that is designed to function as the study's formal object: „Erle'''s'''nis“ (which in German is a pun that combines „Erlebnis“ – adventure experience – and „lesen“ – reading). It is defined as non-propositional knowledge that has been acquired in an individual reading process. Writing about one's own ''Erlesnis'' in new ways is being tried out in essays on ''Traveling on One Leg'' (1989) by Herta Mueller, ''Save the Reaper'' (1998) by Alice Munro, ''Alfred and Emily'' (2008) by Doris Lessing and ''rein GOLD'' (2013) by Elfriede Jelinek respectively. Ideally, an ''Erlesnis'' is based on a text's literariness (see Ulrike Draesner's re-reading of ''Felix Krull'', Thomas Mann's last novel). On de.wikipedia.org an experiment is conducted to find out what community members think about the idea of including, in entries on fiction, sections specifically designed to report about what people felt like when reading a certain text. Finally, a draft typology of ''Erlesnis''-writing is suggested. This contribution is the first of its kind internationally to deal with Wikipedia from the point of view of transfer of knowledge on literature to the public. For the theory of this field of research some new aspects are offered for debate.

'''Keywords:''' ''Alfred and Emily''; Alice Munro; ''Confessions of Felix Krull''; Criticism; Doris Lessing; Elfriede Jelinek; Encyclopedic article; Erlesnis; Fiction; Hans Ulrich Gumbrecht; Herta Müller; Ina Hartwig; Knowledge; Non-propositional knowledge; Open Knowledge; Perlentaucher.de; ''Portraits of a Marriage''; Reading; ''Rein Gold''; Sándor Márai; ''Save the Reaper''; Thomas Mann; Transferring knowledge on literature to the public; ''Traveling on One Leg''; Ulrike Draesner; Wikipedia</text>
      <sha1>19ht12x6bsby9ltpsxwmx300wadc96g</sha1>
    </revision>
  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2810643</id>
      <parentid>2810464</parentid>
      <timestamp>2026-05-20T18:35:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2810643</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="424l9xs55hyxf6390dokb56ukfr4djb" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

==Applications==
* Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]])
* Bird's Implementation 
:- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]])
:- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]])
:- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]])
:- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]])
* Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]])
* Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260520.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== Functors ===
* Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]])
* Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]])
* Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]])

=== Applicatives ===
* Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]])
* Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]])
* Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]])
* Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]])

=== Monads I : Background ===
* Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]])
* Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]])
* Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]])
* Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]])
* IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]])
* Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]])

=== Monads II : State Transformer Monads ===
* State Transformer 
: - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]])
: - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]])
: - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]])

* State Monad 
: - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]])
: - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) 
: - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]])

=== Monads III : Mutable State Monads ===
* Mutability Background 
: - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]])
: - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]])
: - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]])
: - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]])
: - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]])

* Mutable Objects
: - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]])
: - Mutable Data Structures  ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]])

* IO Monad 
: - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]])
: - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]])  
: - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]])  

* ST Monad 
: - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]])
: - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]])
: - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]])

=== Monads IV : Reader and Writer Monads ===
* Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]])
* Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]])
* MonadState Class
:: - State &amp; StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]])
* MonadReader Class
:: - Reader &amp; ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]])
* Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]])

=== Monoid ===
* Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]])

=== Arrow ===
* Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://learnyouahaskell.com/introduction Learn you Haskell]
* [http://book.realworldhaskell.org/read/ Real World Haskell]
* [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material]


[[Category:Haskell|programming in plain view]]</text>
      <sha1>424l9xs55hyxf6390dokb56ukfr4djb</sha1>
    </revision>
  </page>
  <page>
    <title>Luxembourgish language</title>
    <ns>0</ns>
    <id>204724</id>
    <revision>
      <id>2810613</id>
      <parentid>1965661</parentid>
      <timestamp>2026-05-20T15:02:07Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>project box(es)</comment>
      <origin>2810613</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10262" sha1="2jimr1zwo6k0wo7db09pw35iokhnyfq" xml:space="preserve">{{languages}}
{{TOCright}}

== Let’s learn Luxembourgish! ==

This page is supposed to help you learn Luxembourgish. It is less about teaching Luxembourgish, but about learning it. So it is the learners who should come up with the terms or expressions they need, in order to get a translation. 
The idea is to create an interactive dictionary, to improve and expand the vocabulary. This dictionary should be mainly composed of images and audio files, to enable people with a different language background or reduced literacy skills to use it as well. As every dictionary, this is just a tool to help you increase your vocabulary. There will be no grammar or syntax lessons! Nevertheless, this site will provide you with the words you need to train the latter with the help of your teacher.

=== This is how it works ===

As mentioned above, it is the learners who should come up with the terms and expressions they need, since the learning must be meaningful to them. They should take pictures of the objects they want to name in Luxembourgish, or they should record chunks of language or expressions, they want to have translated and upload them. Preferably, they should ask a proficient Luxembourgish speaker to provide the translation, record it and upload it onto this page, to make sure it is done when they need it. 

To keep it as simple as possible, the terms are not organized alphabetically, but according to topics. For every topic, there is a table. In the first column you can find the images, and the following columns are for the respective languages, containing the audio files.

In case you need the translation of an expression, but you do not know anyone to translate it, make sure you add a translation of the expression in either English, French or German, beside your mother tongue, so that someone from Luxembourg can translate it for you.

===== Everyday life / Alldag =====
{| class=&quot;wikitable&quot;
!Farsi
!English
!Lëtzebuergesch
!Français
|-
|[[File:1 Salam....ogg|thumb|Salam]]
|Hello
|[[File:Moien....ogg|thumb|Moien]]
|Salut
|-
|[[File:2 sub bakhair....ogg|thumb|sub bakhair]]
|Good morning
|[[File:Gudde Moien....ogg|thumb|Gudde Moien]]
|
|-
|[[File:3chitor hasty.ogg|thumb|chitor hasty]]
|How are you?
|[[File:Wéi geet et....ogg|thumb|Wéi geet et?]]
|
|-
|[[File:4 khwab khob dashte....ogg|thumb|khwab khob dashte]]
|Did you sleep well?
|[[File:Hues de gutt geschlof....ogg|thumb|Hues de gutt geschlof?]]
|
|-
|[[File:5 subhana....ogg|thumb|subhana]]
|Breakfast
|[[File:Moiesiessen....ogg|thumb|Moiesiessen]]
|
|-
|[[File:6 eshtehai khob....ogg|thumb|eshtehai khob]]
|Enjoy your meal
|[[File:Gudden Appetit....ogg|thumb|Gudden Appetit]]
|
|-
|[[File:7 bad az zohorbakhair....ogg|thumb|bad az zohorbakhair]]
|Good afternoon
|[[File:Gudde Mëtteg....ogg|thumb|Gudde Mëtteg]]
|
|-
|[[File:8 aser bakhair....ogg|thumb|aser bakhair]]
|Good evening
|[[File:Gudden Owend....ogg|thumb|Gudden Owend]]
|
|-
|[[File:9 shab bakhair....ogg|thumb|shab bakair]]
|Good night
|[[File:Gutt Nuecht....ogg|thumb|Gutt Nuecht]]
|
|-
|[[File:10 khawabe khob....ogg|thumb|khawabe khob]]
|Sleep well
|[[File:Schlof gutt....ogg|thumb|Schlof gutt]]
|
|-
|[[File:11 khawabe khob bebene....ogg|thumb|khawabe khob bebene]]
|Nice dreams
|[[File:Dreem eppes schéines....ogg|thumb|Dreem eppes Schéines]]
|
|-
|[[File:12 khoda hafiz....ogg|thumb|khoda hafiz]]
|Goodbye
|[[File:Äddi....ogg|thumb|Äddi]]
|
|-
|[[File:13 chi mekone amroz....ogg|thumb|chi mekone amroz]]
|What are you doing today?
|[[File:Wat méchs du haut....ogg|thumb|Wat méchs du haut?]]
|
|-
|[[File:14 koja mere....ogg|thumb|koja mere]]
|Where are you going?
|[[File:Wou geess du hin?....ogg|thumb|Wou geess du hin?]]
|
|-
|[[File:15 chi mekhai anjam bete....ogg|thumb|chi mekhai anjam bete]]
|What do you want to do?
|[[File:Wat wëlls du maachen....ogg|thumb|Wat wëlls du maachen?]]
|
|-
|[[File:16 chitor bod rozet....ogg|thumb|chitor bod rozet]]
|How was your day?
|[[File:Wéi war däin Dag....ogg|thumb|Wéi war däin Dag?]]
|
|-
|[[File:17 chitor ast zendage....ogg|thumb|chitor ast zendage]]
|How is your life?
|[[File:Wéi geet esou am Liewen....ogg|thumb|Wéi geet et esou am Liewen?]]
|
|}

===== An der Schoul / At school =====

{| class=&quot;wikitable&quot;
|-
! Bild !! Lëtzebuergesch !! Português
|-
| 
[[File:Bläistëft.jpg|thumbnail|Bläistëft]]
 || 
[[File:Bläistëft audio.wav|thumbnail|Bläistëft]]
 || 
[[File:Lápis.wav|thumbnail|Lápis]]
|}

===== Mäi Kierper / My body =====

{| class=&quot;wikitable&quot;
|-
! Bild !! Lëtzebuergesch
|-
| 
[[File:Bild A letz.jpg|thumbnail|A]]
 || 
[[File:Toun A letz.wav|thumbnail|A]]
|-
| 
[[File:Mond Bild letz.jpg|thumbnail|Mond]]
 || 
[[File:Mond Toun letz.wav|thumbnail|Mond]]
|-
| 
[[File:Nues Bild letz.jpg|thumbnail|Nues]]
 || 
[[File:Nues Toun letz.wav|thumbnail|Nues]]
|}

===== Doheem / At home =====

{| class=&quot;wikitable&quot;
|-
! Bild !! Lëtzebuergesch
|-
| 
[[File:Auer Bild letz.jpg|thumbnail|Auer]]
 || 
[[File:Auer Toun letz.wav|thumbnail|Auer]]
|-
| 
[[File:Bänk Bild letz.jpg|thumbnail|Bänk]]
 || 
[[File:Bänk Toun letz.wav|thumbnail|Bänk]]
|-
| 
[[File:Biischt Bild letz.jpg|thumbnail|Biischt]]
 || 
[[File:Biischt Toun letz.wav|thumbnail|Biischt]]
|-
| 
[[File:Dier Bild letz.jpg|thumbnail|Dier]]
 || 
[[File:Dier Toun letz.wav|thumbnail|Dier]]
|-
| 
[[File:Eemer Bild letz.jpg|thumbnail|Eemer]]
 || 
[[File:Eemer Toun letz.wav|thumbnail|Eemer]]
|-
| 
[[File:Kachmaschinn Bild letz.jpg|thumbnail|Kachmaschinn]]
 || 
[[File:Kachmaschinn Toun letz.wav|thumbnail|Kachmaschinn]]
|-
| 
[[File:Raclette Bild letz.jpg|thumbnail|Raclette]]
 || 
[[File:Raclette Toun letz.wav|thumbnail|Raclette]]
|-
| 
[[File:See Bild letz.jpg|thumbnail|See]]
 || 
[[File:See Toun letz.wav|thumbnail|See]]
|-
| 
[[File:Trockner Bild letz.jpg|thumbnail|Trockner]]
 || 
[[File:Trockner Toun letz.wav|thumbnail|Trockner]]
|-
| 
[[File:Wäschmaschinn Bild letz.jpg|thumbnail|Wäschmaschinn]]
 || 
[[File:Wäschmaschinn Toun letz.wav|thumbnail|Wäschmaschinn]]
|-
| 

 || 
[[File:Wäschpolver.wav|thumbnail|Wäschpolver]]
|-
| 
[[File:Zëmmerplanz Bild letz.jpg|thumbnail|Zëmmerplanz]]
 ||
[[File:Zëmmerplanz Toun letz.wav|thumbnail|Zëmmerplanz]]
|-
| 
[[File:Auto Bild letz.jpg|thumbnail|Auto]]
 || 
[[File:Auto Toun letz.wav|thumbnail|Auto]]
|-
| 

 || 
[[File:Bild Toun letz.wav|thumbnail|Bild]]
|-
| 
[[File:Botzmëttel Bild letz.jpg|thumbnail|Botzmëttel]]
 || 
[[File:Botzmëttel Toun letz.wav|thumbnail|Botzmëttel]]
|-
| 
[[File:Bréifkëscht Bild letz.jpg|thumbnail|Bréifkëscht]]
 || 
[[File:Bréifkëscht Toun letz.wav|thumbnail|Bréifkëscht]]
|-
| 

 || 
[[File:Buch Toun letz.wav|thumbnail|Buch]]
|-
| 
[[File:Chrëschtbeemchen Bild letz.jpg|thumbnail|Chrëschtbeemchen]]
 || 
[[File:Chrëschtbeemchen Toun letz.wav|thumbnail|Chrëschtbeemchen]]
|-
| 
[[File:Computer Bild letz.jpg|thumbnail|Computer]]
 || 
[[File:Computer Toun letz.wav|thumbnail|Computer]]
|-
| 
[[File:Dësch Bild letz.jpg|thumbnail|Dësch]]
 || 
[[File:Dësch Toun letz.wav|thumbnail|Dësch]]
|-
| 
[[File:Dreckskëscht Bild letz.jpg|thumbnail|Dreckskëscht]]
 || 
[[File:Dreckskëscht Toun letz.wav|thumbnail|Dreckskëscht]]
|-
| 
[[File:Dusch BIld letz.jpg|thumbnail|Dusch]]
 || 
[[File:Dusch Toun letz.wav|thumbnail|Dusch]]
|-
| 
[[File:Feierläscher Bild letz.jpg|thumbnail|Feierläscher]]
 || 
[[File:Feierläscher Toun letz.wav|thumbnail|Feierläscher]]
|-
| 
[[File:Feiermelder.jpg|thumbnail|Feiermelder]]
 || 
[[File:Feiermelder Toun letz.wav|thumbnail|Feiermelder]]
|-
| 

 || 
[[File:Fotoapparat Toun letz.wav|thumbnail|Fotoapparat]]
|-
| 
[[File:Frigo Bild letz.jpg|thumbnail|Frigo]]
 || 
[[File:Frigo Toun letz.wav|thumbnail|Frigo]]
|-
| 
[[File:Gank Bild letz.jpg|thumbnail|Gank]]
 || 
[[File:Gank Toun letz.wav|thumbnail|Gank]]
|-
| 
[[File:Glänner Bild letz.jpg|thumbnail|Glänner]]
 || 
[[File:Glänner Toun letz.wav|thumbnail|Glänner]]
|-
| 
[[File:Kachplack Bild letz.jpg|thumbnail|Kachplack]]
 || 
[[File:Kachplack Toun letz.wav|thumbnail|Kachplack]]
|-
| 
[[File:Kalenner Bild letz.jpg|thumbnail|Kalenner]]
 || 
[[File:Kalenner Toun letz.wav|thumbnail|Kalenner]]
|-
| 
[[File:Kannerbett Bild letz.jpg|thumbnail|Kannerbett]]
 ||
[[File:Kannerbett Toun letz.wav|thumbnail|Kannerbett]]
|-
| 
[[File:Kannerkutsch Bild letz.jpg|thumbnail|Kannerkutsch]]
 || 
[[File:Kannerkutsch Toun letz.wav|thumbnail|Kannerkutsch]]
|-
| 
[[File:Kichen Bild letz.jpg|thumbnail|Kichen]]
 || 
[[File:Kichen Toun letz.wav|thumbnail|Kichen]]
|-
| 
[[File:Kicheschaf Bild letz.jpg|thumbnail|Kicheschaf]]
 || 
[[File:Kicheschaf Toun letz.wav|thumbnail|Kicheschaf]]
|-
| 
[[File:Klensch Bild letz.jpg|thumbnail|Klensch]]
 || 
[[File:Klensch Toun letz.wav|thumbnail|Klensch]]
|-
| 
[[File:Laptop Bild letz.jpg|thumbnail|Laptop]]
 || 
[[File:Laptop Toun letz.wav|thumbnail|Laptop]]
|-
| 
[[File:Lavabo Bild letz.jpg|thumbnail|Lavabo]]
 || 
[[File:Lavabo Toun letz.wav|thumbnail|Lavabo]]
|-
| 
[[File:Luteschalter Bild letz.jpg|thumbnail|Luteschalter]]
 || 
[[File:Luteschalter Toun letz.wav|thumbnail|Luteschalter]]
|-
| 
[[File:Schaf Bild letz.jpg|thumbnail|Schaf]]
 || 
[[File:Schaf Toun letz.wav|thumbnail|Schaf]]
|-
| 
[[File:Schaukelelch Bild letz.jpg|thumbnail|Schaukelelch]]
 || 
[[File:Schaukelelch Toun letz.wav|thumbnail|Schaukelelch]]
|-
| 
[[File:Schlofkummer Bild letz.jpg|thumbnail|Schlofkummer]]
 || 
[[File:Schlofkummer Toun letz.wav|thumbnail|Schlofkummer]]
|-
| 
[[File:Schnéimännchen Bild letz.jpg|thumbnail|Schnéimännchen]]
 || 
[[File:Schnéimännchen Toun letz.wav|thumbnail|Schnéimännchen]]
|-
| 
[[File:Spullsteen Bild letz.jpg|thumbnail|Spullsteen]]
 || 
[[File:Spullsteen Toun letz.wav|thumbnail|Spullsteen]]
|-
| 
[[File:Steckdous Bild letz.jpg|thumbnail|Steckdous]]
 || 
[[File:Steckdous Toun letz.wav|thumbnail|Steckdous]]
|-
| 
[[File:Toilette Bild letz.jpg|thumbnail|Toilette]]
 || 
[[File:Toilette Toun letz.wav|thumbnail|Toilette]]
|-
| 
[[File:Trap Bild letz.jpg|thumbnail|Trap]]
 || 
[[File:Trap Toun letz.wav|thumbnail|Trap]]
|-
| 
[[File:Vullefudder Bild letz.jpg|thumbnail|Vullefudder]]
 || 
[[File:Vullefudder Toun letz.wav|thumbnail|Vullefudder]]
|-
| 

 || 
[[File:Weichspüler Toun letz.wav|thumbnail|Weichspüler]]
|-
|}

== See Also ==

* [[Wikipedia: Luxembourgish language]]

[[Category:Luxembourgish language|*]]</text>
      <sha1>2jimr1zwo6k0wo7db09pw35iokhnyfq</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Bert Niehaus</title>
    <ns>3</ns>
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      <id>2810625</id>
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      <contributor>
        <username>MediaWiki message delivery</username>
        <id>983498</id>
      </contributor>
      <comment>/* You may be eligible to vote in the U4C election */ new section</comment>
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      <text bytes="33507" sha1="00ojxwz2z98pl7zeqlg1hbamlw0k4dj" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Bert Niehaus!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:Insert-signature.png]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
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* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en &lt;kbd&gt;#wikiversity-en&lt;/kbd&gt;].
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You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:27, 23 January 2016 (UTC)&lt;/div&gt;
{{Robelbox/close}}

== Subpages and Links ==

Learning projects are best organized with [[Wikiversity:Subpages|subpages]] (like Wikibooks) rather than as individual pages (like Wikipedia). I've moved the E-Proof Future Development page to [[E-Proof/Future Development and Feature Discussion]].

When linking to Wikiversity pages, use internal links with &lt;nowiki&gt;[[Title]]&lt;/nowiki&gt; or &lt;nowiki&gt;[[Title|Display]]&lt;/nowiki&gt; syntax. When linking to subpages, use &lt;nowiki&gt;[[/Subpage/]]&lt;/nowiki&gt; syntax. See [[Making links]] for more information.

Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:26, 20 December 2016 (UTC)

Thank you Dave for your support

== Risk Management ==

Hi Bert Niehaus!

Your resource [[Risk Management]] appears to be ready for learners! Would you like it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 13:52, 1 May 2017 (UTC)

Thank you for announcement offer, Excuse me for late reply.
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:45, 14 August 2017 (UTC)

== Trust ==

I see you have recently added both [[Trust in Data, Information and Knowledge]] and [[Trust in Capacity Building Material]]. Are you ultimately building a learning project on [[Trust]] that these and any additional pages belong to, or are they part of [[Risk Management]]?

We try to organize Wikiversity content by learning project rather than just learning resource. --
 [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:26, 11 July 2017 (UTC)

(1) First I though I just plan a trust learning resource for risk management, but then I looked on my lecture about encryption, digital signature and number theory it makes sense to me to broaden the mathematical perspective on encryption to application and trust. On the other hand from trust it makes sense to learn about digital signature and encryption. So it makes more sense to build a learning resource on trust. Would be great to have some psychologist on board to approach the trust topic from their angle, but who knows collaborative development of content could trigger wonderful things, 

Thank you for your support, Dave
 [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 16:20, 12 July 2017 (UTC)

== Wikipedia Imports ==

All Wikimedia wikis are licensed using Creative Commons CC-BY-SA licensing. The BY part of that license requires crediting the source whenever licensed content is reused. Within wiki software, the best way to credit the source is by including their edits in the page history. This is done using [[Special:Import]], available to Wikiversity custodians and curators. If you would like to import any additional Wikipedia pages, please either use [[Wikiversity:Import]] to request page imports or see [[Wikiversity:Curators]] to request curatorship so you may process your own imports. Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 12 August 2017 (UTC)

   [[Special:Import]] shows a 'permission error' for me as an ordinary wiki author
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:40, 13 August 2017 (UTC)

:Is the curator status the only option as a user to do branching of content development from Wikipedia to Wikiversity?

::Yes, the only way to import pages from Wikipedia yourself is with curator status. Otherwise, you are welcome to post at [[Wikiversity:Import]] and one of us can import pages for you. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:48, 12 August 2017 (UTC)

== Forking and Tailor Eductional Resources ==
* '''[[/Forking/|Forking of Learning Resources and tailor Learning Resources for Target Groups]]'''
* '''[[Open Educational Resources/Localization]]'''
=== Teachers Informations: Forking Info for Learner Profile ===
Sometime a learning topic can be developed for different educational level. Teacher information can be used to fork from a parent module to submodules tailored for the target group. That is my solution to support branching within the existing IT infrastructure.

See '''[[Water#Information_for_Teachers|Water]]''' learning resource.
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 14:45, 29 September 2017 (UTC)

== Curator Status ==

Based on the support expressed at [[Wikiversity:Candidates for Curatorship/Bert Niehaus]], you are now a curator. Congratulations!

You should see new tools available under [[Special:SpecialPages]] and on the More menu on each page. Use them wisely, and let me know if you have any questions.

Also, please update [[Wikiversity:Support staff]] and add your information.

[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:48, 22 August 2017 (UTC)

Ttank you very much,
Bert
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 04:54, 23 August 2017 (UTC)

== Sustainable Development Goals ==

Hi Bert Niehaus!

Your resource [[Sustainable Development Goals]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 02:47, 29 August 2017 (UTC)

OK thank you, learning tasks are still missing. And I want to insert the SDG logos on pages, but have to read licensing by United Nations before using them in Wikiversity. SDG Logos seem not possible to import from Wikipedia.

:* I've looked at the logos at the Wikipedia [[w:Sustainable Development Goals|Sustainable Development Goals]]. One is at Commons and the other is fair use which is also allowed here by USA copyright law. You are free to upload them here or use the Commons version, where available. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 14:08, 29 August 2017 (UTC)
:* Thank you --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 08:49, 30 August 2017 (UTC)

Logos inserted and started to assign Sustainable Developement Goals to Learning Resources e.g. on [[Water]] and [[Collaborative Mapping]]

== PanDocElectron and Water ==


Hi Bert Niehaus!

Your resources [[PanDocElectron]] and [[Water]] appear to be ready for learners! Would you like to have them announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 12:12, 24 October 2017 (UTC)

Thank you for regarding the learning resource as ready. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 13:46, 24 October 2017 (UTC)

== Commercial Data Harvesting ==

Hi Bert! I love your page on [[Commercial Data Harvesting]] and its 5 basic constituents.

I'm currently writing an article on the subject and was wondering if there's any literature that mentions the concept (and its constituents) like that Thank you, Ana Leonardi

 [[User:Leonardiac|Leonardiac]] ([[User talk:Leonardiac|discuss]] • [[Special:Contributions/Leonardiac|contribs]]) 15:06, 29 October 2017 (UTC)

== 3D Modelling ==

Hi Bert Niehaus!

Your learning resource [[3D Modelling]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:54, 21 January 2018 (UTC)

Yes, fine, thank you for annoucing it. Still working on the 3D modelling, but some learners might benefit from the resource.
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 16:57, 21 January 2018 (UTC)

== Expert Focus Group for Space and Global Health ==

Hi Bert Niehaus!

Your resource [[Expert Focus Group for Space and Global Health]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:30, 5 March 2018 (UTC)

There is lot of work to do, but yes if you think it is worth publishing, it is Ok for me.
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:59, 6 March 2018 (UTC)

== Share your experience and feedback as a Wikimedian in this global survey ==

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== Think globally - act locally ==

Hi Bert Niehaus!

Your resource [[Think globally - act locally]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 03:00, 23 April 2018 (UTC)

Thank you Marshall, It is ok, publishing the resource.
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 05:09, 28 April 2018 (UTC)

== Structuring Data ==

Hi Bert Niehaus!

Your resource [[Structuring Data]] appears well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 03:39, 24 July 2018 (UTC)

Thank you Marshall, I am Ok with publishing the resource --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 07:38, 25 July 2018 (UTC)

== Humanitarian Open Streetmap ==

Hi Bert Niehaus!

Your resource [[Humanitarian Open Streetmap]] appears to be well-developed and ready for learners and participants! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:39, 28 August 2018 (UTC)

Thank you Marshallsumter, I am Ok with publishing the resource
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 06:20, 2 September 2018 (UTC)

== Collaborative mapping ==

Hi Bert Niehaus!

Your resource [[Collaborative mapping]] appears well-developed and ready for learners and participants! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:35, 24 September 2018 (UTC)

Thank you Marshall, I am OK with annoucement,
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 09:59, 26 November 2018 (UTC)

== Moving Average ==

Hi Bert Niehaus!

Your statistics resource [[Moving Average]] is well-developed and appears ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 02:40, 10 December 2018 (UTC)

Dear Marshall,
currently Moving Average needs some learning tasks to be added. Sorry for late reply. 
Best regards,
bert
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 18:01, 21 February 2019 (UTC)
:: Dear [[User:Marshallsumter|Marshallsumter]] the learning resource Moving Average can be annouced on Main Page News -

== Systems Thinking ==

Hi Bert Niehaus!

Your landing page [[Systems Thinking]] appears well-developed and ready to assist learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:13, 18 February 2019 (UTC)

Dear Marshall, I need a bit of work for systems thinking to be ready. I would recommend that is NOT announced on the Main page. Will come back to you, according to this learning  module. Thank you, Bert
--[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 18:00, 21 February 2019 (UTC)

== Community Insights Survey ==

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The Wikimedia Foundation is asking for your feedback in a survey about your experience with {{SITENAME}} and Wikimedia. The purpose of this survey is to learn how well the Foundation is supporting your work on wiki and how we can change or improve things in the future. The opinions you share will directly affect the current and future work of the Wikimedia Foundation.

Please take 15 to 25 minutes  to '''[https://wikimedia.qualtrics.com/jfe/form/SV_0pSrrkJAKVRXPpj?Target=CI2019List(other,act5) give your feedback through this survey]'''.  It is available in various languages.

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Sincerely,
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Sincerely,
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There are only a few weeks left to take the Community Insights Survey! We are 30% towards our goal for participation. If you have not already taken the survey, you can help us reach our goal!
With this poll, the Wikimedia Foundation gathers feedback on how well we support your work on wiki. It only takes 15-25 minutes to complete, and it has a direct impact on the support we provide.

Please take 15 to 25 minutes  to '''[https://wikimedia.qualtrics.com/jfe/form/SV_0pSrrkJAKVRXPpj?Target=CI2019List(other,act5) give your feedback through this survey]'''.  It is available in various languages.

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Sincerely,
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==A barnstar for you==
{| style=&quot;border: 1px solid {{{border|gray}}}; background-color: {{{color|#fdffe7}}};&quot;
|rowspan=&quot;2&quot; style=&quot;vertical-align:middle;&quot; | {{SAFESUBST:&lt;noinclude /&gt;#ifeq:{{{2}}}|alt|[[File:Kindness Barnstar Hires.png|100px]]|[[File:Random Acts of Kindness Barnstar.png|100px]]}}
|rowspan=&quot;2&quot; |
|style=&quot;font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;&quot; | '''The Random Acts of Kindness Barnstar'''
|-
|style=&quot;vertical-align: middle; border-top: 1px solid gray;&quot; | {{{Thank you for supporting my proposal on WikiQuiz}}}
|}
[[User:RIT RAJARSHI|RIT RAJARSHI]] ([[User talk:RIT RAJARSHI|discuss]] • [[Special:Contributions/RIT RAJARSHI|contribs]]) 16:27, 2 April 2020 (UTC)

== SIR model and COVID ==
Thank you for the suggestion on my page ([[COVID-19/Iluvalar]]) , I think you misunderstood the object of that chapter completely. The point is to explain that the first exponential part of the data, can only be the test production curve and have nothing to do with the actual virus. The starting data and the production rate of the virus test had it own value and any contact with the real infection curve of the virus could only be accidental and highly unlikely. For the actual SIR model I used, see the line &quot;model apr 4&quot; and &quot;model apr 7&quot; later on the page, I'll try to add the data for the following month, but I think I was pretty accurate with that. [[User:Iluvalar|Iluvalar]] ([[User talk:Iluvalar|discuss]] • [[Special:Contributions/Iluvalar|contribs]]) 01:45, 10 May 2020 (UTC)

: {{At|Iluvalar}} Agree with you, testing is not performed in a randomly selected cohort of citizens and testing is not driven by a controlled study design. Asymptomatic patients are not tested and even symptomatic patients were not tested because staff in health care facilities did not decide to test the patient or  decided to test other patients with higher priority first, because of the limited test capacity. Shall we add a specific submodul to specify the limitation of modelling due to the limitation of dada aquisitions. Or do you have other suggestions for learning resource? --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:26, 11 May 2020 (UTC)

== Your class on Commercial Data Harvesting ==

Hi {{PAGENAME}}:

Since I discovered it I have been following your class on [[Commercial Data Harvesting]] with interest. I am fairly new at WV and still have not figured out what users like me who are not lecturers at WV are allowed to do. At Universities that  attended in person as a student, free exchange of information was tolerated if not encouraged.

Anyway, what I came here to suggest is that the course assumption that '''...users regard themselves as ''customer'' of an provider of a free digital service''', may have been accurate in the past at some [[w:Big tech]] offerings, in some countries, but it is more questionable if all users of such services have always considered themselves as a customer of a free service.
: Thank you for your feedback, added a comment according to your feedback for the learning resoruce --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 11:56, 29 November 2020 (UTC) 

Take for example the user revolt reffered to in [[w:Reddit#Company_history]], hardly the action of individuals gratefull for a free service they are receiving? 

I hope I am making sense, contributing positively to WV, and not wasting your time unnecessarily. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 19:30, 28 November 2020 (UTC)
== Files Missing Information ==
Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information.
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* [[:File:Data analysis digital learning environments.svg]]
{{colend}}
[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 00:25, 9 January 2022 (UTC)
:: DONE --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 09:48, 8 February 2022 (UTC)

== Files Missing Information ==
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* [[:File:Eye image smoother.png]]
* [[:File:Eye image.png]]
{{colend}}
[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 19:45, 7 February 2022 (UTC)

:: DONE --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 09:48, 8 February 2022 (UTC)
== Files Missing Information ==
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[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 05:15, 23 December 2022 (UTC)

== Help debugging my image map ==

I have attempted to create an image map at: [[The Wise Path]] I followed the (Wikipedia) image map example. The map does not work. I will appreciate your help in debugging my image map. Thanks! [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:49, 23 May 2023 (UTC)

:Excuse me for the late reply [[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 10:04, 25 May 2023 (UTC)

: I fixed it! thanks

== Lightboard ==
FYI, I marked [[Lightboard]] for proposed deletion since it is empty. Perhaps you can add a sentence or two and a couple of good links to make the page worthwhile? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:01, 3 October 2024 (UTC)

== Complex Analysis/Curves ==
In [[Complex Analysis/Curves]], I find edits by [[User:Eshaa2024]] for which I cannot quickly confirm are good. You are the author of the page, so you may want to have a look. Perhaps Eshaa2024 is your collaborator; I don't know. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:16, 12 December 2024 (UTC)

== You may be eligible to vote in the U4C election ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
I am contacting you because you previously voted in elections related to the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee (U4C)]]. You may be eligible to vote in the current U4C election, which is open now and closes on 2 June 2026. You can find out more about the candidates and the election on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|the election page on Meta]], and from there you can access the vote itself. Your participation in these elections is important to the governance of Wikimedia communities, and your time spent learning about the candidates and voting is appreciated.

-- In cooperation with the U4C, [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]])&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:Keegan (WMF)|Keegan (WMF)]] ([[m:User_talk:Keegan (WMF)|talk]]) 17:20, 20 May 2026 (UTC)
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      <sha1>00ojxwz2z98pl7zeqlg1hbamlw0k4dj</sha1>
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  <page>
    <title>Layers of the Atmosphere</title>
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      <text bytes="4014" sha1="03np0oscfqabhmdcrbo6puirve9u2jx" xml:space="preserve">{{environmental science}}
[[File:Layers of the atmosphere.PNG|thumb|Diagram of the layers of the Atmosphere]]
There are four layers in our atmosphere, each separated by temperature changes. These layers are:
*'''Troposphere'''
*'''Stratosphere'''
*'''Mesosphere'''
*'''Thermosphere'''
These 4 layers each have unique characteristics to each one of them. In this lesson, we will learn these layers one by one.

==Troposphere==
'''Distance''': &quot;The troposphere starts at Earth's surface and goes up to a height of '''7 to 20 km''' (4 to 12 miles, or 23,000 to 65,000 feet) above sea level&quot;&lt;sup&gt;[http://www.windows2universe.org/earth/Atmosphere/troposphere.html [1]]&lt;/sup&gt;
[[File:Troposphere CIMG1853.JPG|thumb|left|A photo of the Troposphere]]
This layer is the lowest layer of the Earth, the ''troposphere'' (trop-os-feer). This layer is where we, the humans, live in (and all the weather is at!). This layer contains nearly 90% of the atmosphere's total mass! Almost all of the Earth's water vapor, carbon dioxide, air pollution, clouds, weather and life forms live in. The word, &quot;troposphere&quot;, literally means &quot;change/turning ball&quot;, as the gases turn and mix in this layer. These gases mix due to the differences in air temperature and density.

==Stratosphere==
'''Distance''': &quot;The bottom of the stratosphere is around '''10 km''' (6.2 miles or about 33,000 feet) above the ground at middle latitudes. The top of the stratosphere occurs at an altitude of '''50 km''' ('''31 miles''')&quot;&lt;sup&gt;[https://scied.ucar.edu/shortcontent/stratosphere-overview [2]]&lt;/sup&gt;
[[File:Ozone 2001sept17 lrg.jpg|thumb|right|Hole in the Ozone Layer]]
The layer above the Troposphere is known as the '''Stratosphere''' (strat-os-feer). Gases in the stratosphere are layered (why it has the prefix, &quot;strato&quot;, meaning layered), the air is very thin (little moisture as well), and is extremely cold (lower stratosphere). Although the lower part of the stratosphere is cold, the heat in the stratosphere increases as altitude increases. This is because of the ozone layer in the upper part of the Stratosphere. The Ozone layer absorbs ultraviolet radiation from the sun, and as a result, warms up the air. This ozone layer is key to our safe living on Earth.

==Mesosphere==
'''Distance''': &quot;The mesosphere starts at '''50 km''' ('''31 miles''') above Earth's surface and goes up to '''85 km''' ('''53 miles''') high&quot;&lt;sup&gt;[http://www.windows2universe.org/earth/Atmosphere/mesosphere.html [3]]&lt;/sup&gt;

Above the Stratosphere is the '''Mesosphere''' (mez-os-feer), the middle layer of the atmosphere (''meso-'' meaning &quot;middle&quot;). This layer is the coldest layer (temperature decreases as altitude increases). Temperatures can be as low as -93 degrees Celsius at the top of the Mesosphere.

==Thermosphere==
'''Distance''': &quot;It extends from '''about 90 km''' ('''56 miles''') to between '''500 and 1,000 km (311 to 621 miles)''' above our planet&quot;&lt;sup&gt;[http://www.windows2universe.org/earth/Atmosphere/thermosphere.html [4]]&lt;/sup&gt;
[[File:Sunset from the ISS.JPG|thumb|left|The thermosphere is included in this picture.]]
The uppermost atmosphere is called the '''Thermosphere''' (therm-os-feer). Here, the temperature again increases with altitude. This is because atoms of nitrogen and oxygen absorb high-energy solar radiation and give off thermal energy... this causes the temperature to increase up to 1,000 degrees Celsius. 

Even though it may seem, it is actually not hot in the Thermosphere. Pay close attention to these details:
*Temperature is different from heat
**'''Temperature''' is the measure of the average energy of particles in motion. Thus, the thermosphere has particles moving very fast.
**'''Heat''' is the transfer of thermal energy between objects of different temperatures. Therefore, particles must touch one and another to transfer heat.

In the Thermosphere, it is has a low density--thus the particles in the Thermosphere usually don't collide, thus not giving off heat.

[[Category:Atmospheric science]]</text>
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  <page>
    <title>Forex Trading</title>
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        <username>~2026-30178-08</username>
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      <comment>/* Further reading */</comment>
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      <comment>Reverted edit by [[Special:Contributions/~2026-30178-08|~2026-30178-08]] ([[User_talk:~2026-30178-08|talk]]) to last version by [[User:XXBlackburnXx|XXBlackburnXx]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="2439" sha1="18vlx7lif1hv1bih6jtcl9g76q8h70e" xml:space="preserve">Forex is a short hand name for Foreign Exchange (also known as the FX Market), which is a type of financial market. Forex Trading is whereby an institute(s) or individual sell and buy one country's currency for another country's currency for an intended purposes. The Forex market has been around for long time but it's popularity has been made possible by the internet in the early nineties. This introduced a Forex market to electronic platform which made trading currencies faster and efficient. The introduction of electronic platform meant that Forex trading could be done through one's computer through the browser by web based trading applications or install-able trading software provided by the broker or banks. This lead to more and more of individual traders as to compared to before where there were few individual and who had to call their broker or be on the trading floor in order to trade currencies. Today the Forex is considered the most '''largest and liquid market in the world''', and almost $3 trillion is traded everyday and in a certain day the market can trades as much as $7 trillion per day. 

Even though Forex market is the largest financial market in the world, the market is ''decentralized'' or over-the-counter (OTC).

==Further reading==
* Africa in the news: Nigeria’s forex market, Tanzania’s falsely credentialed civil servants, and WEF Africa news
* [http://scalar.usc.edu/works/farnham-research/top-5-forex-risks-traders-should-consider Top 5 Forex Risks Traders Should Consider]
* [http://jsmith.cis.byuh.edu/books/policy-and-theory-of-international-economics/s18-01-the-forex-participants-and-obj.html The Forex: Participants and Objectives]

===Books===
* [http://jsmith.cis.byuh.edu/books/policy-and-theory-of-international-economics/index.html Policy and Theory of International Economics]

==The Basic of Foreign Exchange Trading==
* {{cite book |last= Smith |first= Courtney |date= 2010 |title= [[How to Make a Living Trading Foreign Exchange : A Guaranteed income for life]] |url= |location= Hoboken, New Jersey. Published simultaneously in Canada. |publisher= John Wiley &amp; Sons, Inc.|isbn= 978-0-470-44229-6 |author-link= }}
* {{cite book |last= Dolan |first= Brian |date= 2011 |title= Currency Trading For Dummies, 2nd edition |url= |location= Indianapolis, Indiana. Published simultaneously in Canada  |publisher= Wiley Publishing, Inc. |page= |isbn= 978-1-118-01851-4 |author-link= }}</text>
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  <page>
    <title>Social Victorians/People/Paget Family</title>
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      <text bytes="30982" sha1="rigvdjaoonb4wps1e9xjghvwtu33euv" xml:space="preserve">== Overview ==
=== Arthur Henry Fitzroy Paget ===

* “Possessed of considerable wealth, Paget won renown less as a soldier than as a bon vivant; his close friendship from boyhood with the prince of Wales (later King Edward VII) — the two were regular social companions — was an incalculable asset throughout his career. Pompous and verbose in speech, headstrong when aroused, though gallant in the field, Paget was neither cool nor clear-headed; intellectually shallow, he boasted to have ‘lived history rather than read it’ (''Times'' obit.).”&lt;ref name=&quot;:9&quot;&gt;White, Lawrence William. “Paget, Sir Arthur Henry Fitzroy.” ''Dictionary of Irish Biography''. DOI: https://doi.org/10.3318/dib.007159.v1.&lt;/ref&gt;
* ”After retiring in 1918 he spent most of his time in Cannes, prominent in yachting circles on the Riviera. His other recreations included racing, hunting, fishing, and golf; an avid gardener, he had a wide amateur knowledge of botany.”&lt;ref name=&quot;:9&quot; /&gt;
* ”Paget died in Cannes on 9 December 1928, leaving an unsettled estate of £22,708.”&lt;ref name=&quot;:9&quot; /&gt;

=== Minnie Paget (Mrs. Arthur Paget) ===

* “… of New York, USA; strong-minded and vivacious, she became a prominent London hostess” after her wedding to Arthur Paget.&lt;ref name=&quot;:9&quot; /&gt;
* Minnie Paget attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]]. She was one of 2 women dressed as Cleopatra.
* Fiske

=== Almeric Fitzroy ===
* Almeric Fitzroy doesn't really belong on this page. The very big generation of Pagets has men with the names Almeric and Fitzroy, so he ended up here for now.
* The 7th son of a landed general, a subsequent son of a subsequent son, Almeric Fitzroy was a civil servant. David Cannadine describes him as an example of &quot;landed-establishment life&quot; and one of the &quot;genteel mandarins&quot;:&lt;blockquote&gt;He was a great grandson of the third Duke of Grafton, and his mother was a daughter of Lord Feversham. He began his official life as an Inspector of Schools in the Education Department of the Privy Council. The appointment was arranged by family influence, and it gave Fitzroy time to hunt three days in every fortnight.... In 1884, [[Social Victorians/People/Carlingford|Lord Carlingford]] transferred him to the Privy Council Office itself; in 1895 the Duke of Devonshire (who had just become Lord President) made him his private secretary; and three years later, the combination of family influence and the Duke's patronage brought him the Clerkship of the Privy Council, which he held until his retirement in 1923. Throughout this period, he was on the closest terms with the leading politicians of the day, he moved easily in royal and patrician society, he was a well-known figure in the clubs of London, and he spent many a weekend at Chatsworth, Lissadell, Osterley, Longleat, and Euston.&lt;ref name=&quot;:8&quot;&gt;Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.&lt;/ref&gt;{{rp|242}}&lt;/blockquote&gt;

* Also, according to Cannadine, &quot;Almeric Fitzroy wrote books about his ancestors, and was a trustee of the Duke of Grafton's settlement.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|242}}

== Also Known As ==
*Family name: Paget
*Sir Arthur Fitzroy and Mrs. Minnie (Mary Stevens) Paget
**General Rt. Hon. Sir Arthur Henry Fitzroy Paget
*Mr. and Mrs. Cecil Paget
*Mr. and Mrs. George Ernest Paget
**and Miss Hylda Paget
*Mr. Gerald and Mrs. Lucy Paget
*Almeric Fitzroy
**Mr. Fitzroy (nom de plume)

== Acquaintances, Friends and Enemies ==

== Timeline ==
'''1877 January 2''', Gerald Cecil Stewart Paget and Lucy Annie Emily Gardner married.&lt;ref name=&quot;:6&quot;&gt;&quot;Lucy Annie Emily Gardner.&quot; {{Cite web|url=https://www.thepeerage.com/p4699.htm#i46982|title=Person Page|website=www.thepeerage.com|access-date=2021-12-01}} https://www.thepeerage.com/p4699.htm#i46982.&lt;/ref&gt;

'''1878 July 27''', Mary Stevens and Arthur Henry Fitzroy Paget married.&lt;ref name=&quot;:3&quot;&gt;&quot;Mary Stevens.&quot; {{Cite web|url=https://www.thepeerage.com/p3392.htm#i33914|title=Person Page|website=www.thepeerage.com|access-date=2020-10-18}}&lt;/ref&gt;

'''1889 June 17''', Alexandra Harriet Paget and [[Social Victorians/People/Colebrooke|Edward Arthur Colebrooke]] married.&lt;ref&gt;&quot;Alexandra Harriet Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p880.htm#i8792|title=Person Page|website=www.thepeerage.com|access-date=2020-12-13}}&lt;/ref&gt; (They attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House and are treated on the [[Social Victorians/People/Colebrooke|Colbrooke page]].)

'''1897 July 2''', Mr. Arthur and Mrs. Minnie Paget attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did his brother Gerald Cecil Stewart Paget.

'''1902''', Arthur Henry Fitzroy Paget was promoted to the rank of General.&lt;ref name=&quot;:2&quot;&gt;&quot;General Rt. Hon. Sir Arthur Henry Fitzroy Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p3392.htm#i33913|title=Person Page|website=www.thepeerage.com|access-date=2020-10-18}}&lt;/ref&gt;

'''1914 March''', Arthur Henry Fitzroy Paget “precipitated the so-called Curragh ‘mutiny’ [in Ireland] by grossly and melodramatically misrepresenting orders regarding precautionary troop movements intended to safeguard against possible seizures of arms depots by the Ulster Volunteers.”&lt;ref name=&quot;:9&quot; /&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
[[File:Клеопатра VII.jpg|thumb|359x359px|Black basalt Egyptian statue of Cleopatra as an Egyptian goddess or Arsinoe II]]

=== Minnie Paget ===
[[File:Mary-Minnie-ne-Stevens-Lady-Paget-as-Cleopatra.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed as Egyptian Cleopatra in an historical costume with fans and a very ornate head-dress|&quot;Minnie,&quot; Lady Paget as Cleopatra. ©National Portrait Gallery, London.|left]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Minnie Paget, Mrs. Arthur Paget, walked in the &quot;Oriental&quot; procession as Cleopatra.&lt;ref name=&quot;:0&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref name=&quot;:4&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; 

John Thomson's portrait (left) of &quot;Mary ('Minnie', née Stevens), Lady Paget as Cleopatra&quot; in costume is photogravure #145 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|album presented to the Duchess of Devonshire]], one copy of which is in the National Portrait Gallery.&lt;ref name=&quot;:1&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Mrs. Arthur Paget as Cleopatra.&quot;&lt;ref&gt;&quot;Mrs. Arthur Paget as Cleopatra.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158508/Mary-Minnie-ne-Stevens-Lady-Paget-as-Cleopatra.&lt;/ref&gt;

The Lafayette Negative Archive has Minnie Paget in another pose in this same costume:

* http://lafayette.org.uk/pag1400a.html

The photograph in the album was taken by John Thomson, but the one in the Lafayette Negative Archive has to have been taken by Lafayette, probably in its studio.

The headdress looks better in the portrait published in the album, which also shows the large plume fan behind her that was, presumably, carried by an attendant as she processed.

The Egyptian statue (right) of Cleopatra as an Egyptian goddess (or of Arsinoe II) is made of black basalt and dates from the second half of the 1st century B.C.E.&lt;ref&gt;&quot;Statue of Cleopatra.&quot; Hermitage Museum, St. Petersburg, Russia. Wikimedia Commons (retrieved August 2025), filename Клеопатра VII.jpg https://commons.wikimedia.org/wiki/File:%D0%9A%D0%BB%D0%B5%D0%BE%D0%BF%D0%B0%D1%82%D1%80%D0%B0_VII.jpg.&lt;/ref&gt; It is now in the collection at the Hermitage Museum in St. Petersburg, Russia.

==== Newspaper Reports ====
Minnie Paget apparently had a Black attendant, which the ''Morning Post'' described using the n-word, quoted below; offensive language appears in other reports as well, like the description of her appearance in the report of the American ''Providence [Rhode Island] Evening Telegram''.
*She was dressed in an &quot;Egyptian costume, the train of black crepe de chine embroidered with gold scarabaeus and lined with cloth of gold; skirt of black gauze with lotus flowers worked in gold, and sash of gauze tissue wrought with stones and scarabaeus. The bodice, glittering with gold and diamonds, was held up on the shoulders with straps of large emeralds and diamonds. The square head-dress was of Egyptian cloth of gold, the sphinx-like side pieces being striped black and gold encrusted with diamonds, and in the middle of the forehead hung a large pearl from a ruby; above was the ibis with outstretched wings of diamonds and sapphires, and beyond were peacock feathers standing out, and the back was all looped with pearls and amber. The remainder of the head-dress was of uncut rubies and emeralds, all real stones, surmounted by the jewelled crown of Egypt; round the neck were row upon row of necklaces of various gems, reaching to the waist, and a jewelled girdle fell to the hem; a nigger [sic] held a fan of ostrich feathers over her head.&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}}
*&quot;Mrs. Arthur Paget appeared in an Egyptian costume, the train being of black crepe de chine embroidered with gold scarabæns [sic], and lined with cloth of gold; skirt of black gauze with lotus flowers worked in gold, and sash of gauze tissue wrought with stones and scarabæns. The bodice, glittering with gold and diamonds, was held up on the shoulders with straps of large emeralds and diamonds.&quot;&lt;ref name=&quot;:5&quot;&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 4 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 4, Col. 3b}}
*&quot;Another Cleopatra was Mrs. Arthur Paget, who really looked the character, as she is so dark and Oriental in appearance. Mrs. Paget had a black attendant.&quot;&lt;ref&gt;&quot;Gorgeous Affair. Costume Ball Given by the Duchess of Devonshire in London Last Evening. Many Americans Present. Duchess of Marlborough Appeared as ‘Columbia’ and Depew as Washington.&quot; ''Providence [Rhode Island] Evening Telegram'' Saturday 3 July 1897: 9, Col. 3b [of 8]. ''Google Books''. Retrieved September 2023. https://books.google.com/books?id=gvJeAAAAIBAJ.&lt;/ref&gt;
*&quot;There were also two Cleopatras ..., and Mrs. Arthur Paget looked her character to the life, and her jewels were quite the most magnificent in the room. Mr. Gerald Paget walked beside her, attired very effectively as Mark Antony.&quot;&lt;ref name=&quot;:7&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 32, Col. 2c}}
[[File:Queen Victoria (1887).jpg|thumb|Queen Victoria wearing the Coronation Necklace and Earrings, small-Diamond Crown and Koh-i-Noor brooch.]]
==== Commentary on Her Costume ====
[[File:Grand sphinx de Tanis - Musée du Louvre Antiquités égyptiennes N 23 ; A 23 ; Salt 3837 - photo 2.jpg|thumb|Sphinx Headdress with Horizontal Stripes, Musée du Louvre Antiquités Égyptiennes]]This complex costume has some Egyptian-looking elements, but it is mostly Victorian in style and line. The Victorian line is present in the corseted waist, the crown part of the headdress and the train. The frou-frou and much of the jewelry reflect a Victorian sense of style. Minnie Paget's waist in this portrait is strikingly small and has an improbably smooth curve, and the photograph looks like it has been retouched.
* The ''Morning Post'' says that Paget's headdress is &quot;surmounted by the jewelled crown of Egypt,&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}} but the little crown on top of her head looks more like Queen Victoria's small diamond mourning crown (right)&lt;ref&gt;{{Cite journal|date=2025-03-12|title=Small Diamond Crown of Queen Victoria|url=https://en.wikipedia.org/w/index.php?title=Small_Diamond_Crown_of_Queen_Victoria&amp;oldid=1280094126|journal=Wikipedia|language=en}}&lt;/ref&gt; than any of the historical crowns of Egypt. The design is different, but the size is about the same. Paget's headdress has so many jewels on it, it looks more like a European helmet than anything worn by Egyptian royalty. The pearl dangling between her eyes from the filet or crown would have been very distracting to wear.
* The striped lappets, which stick out from the sides of that helmet, evoke the headdress of the sphinx, several versions of which exist in good condition in museums in Europe (Sphinx in the Louvre, below right).
* Like the lappets, the decorations on Paget's costume hint at Egypt, but the costume is essentially Victorian.
* Paget's [[Social Victorians/Terminology#Corset|foundation garments]] are clearly Victorian, which explains the tiny waist (even in the image that is not retouched) and the emphasis on bosom and hips. And they explain why this costume looks so very Victorian and so not Egyptian.
* This visually busy costume is covered in Victorian [[Social Victorians/Terminology#Frou-frou|frou-frou]]: elaborate and varied decorations with jewels, layers, repeated motifs and different fabrics and trim. The fabric worn by the historical Egyptians would have been light, sometimes almost transparent. These Victorians were wearing silk. which was used to make the velvets and satins. Made of synthetic fibers, our velvets and silks would be too warm for a July party, but silk fabrics breathe, drape better and are much lighter and richer. What was supposed to make the costume Egyptian are the stereotypical motifs associated with Egypt like ibis wings on the bodice and train, the horizontal stripes motif, the arm bands and multiple bracelets, the sphinxy headdress and so on.
* This costume must have been heavy, with all its layers, metal and jewels.
* The bodice — &quot;glittering with gold and diamonds&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}} — is breast-plate- or cuirass-like with rounded &quot;scales.&quot; The shoulder straps of the bodice, decorated with diamonds and very large emeralds, are mentioned in the newspaper reports. The &quot;scales&quot; are thick, stiffened, framed or outlined with something possibly metallic like a textured braid or cord, attached at the top and free at the bottom. Besides the &quot;scales&quot; the bodice is heavily ornamented with huge jeweled pieces at the top and bottom. The top piece is a complex mixture of ibis wings made of jewels and precious-metal sequins with a large brooch-like flower and a pendant diamond-encircled cabochon. The bottom piece nearest the bodice is on top of the girdle 
* Like the busyness, the layers are typical of Victorian design. Distinguishing among the layers is very difficult because of the way the costume was constructed and because the photograph is in black and white. In particular, we can't tell how many layers there are or what's under what.[[File:Sarah_Bernhardt_1844-1923,_fransk_skådespelare_-_1890_SLSA_1270_34_foto_186.jpg|thumb|Sarah Bernhardt as Cleopatra, '''1890, 1891?''', Sarony, NY]]
** Although we can't see them, the foundation garments are essential to the layering of this costume.
** Jeweled or beaded elements are on top, but what they are mounted on is not clear.
** One layer is the distinctive overskirt that is not solid fabric — it might be strings with beads or jewels attached, or open-weave netting or ribbons, hinting perhaps that Cleopatra would not have been wearing as much clothing as Paget is. 
** Draped around her hips and falling down the front is a striped girdle or sash. The girdle is stitched in folds at the hips and falls straight down with what would be the ends of a tied girdle gradually widening. Stitching the girdle down — rather than tying it — controls the bulkiness around the waist and hips and keeps the &quot;ends&quot; stiff and in place.
** The girdle acts as a frame to the vertically striped ornament decorating the front of the skirt.
** The portrait of Sarah Bernhardt (right) shows her costumed as Cleopatra but very differently from Paget. She does not seem to be wearing Victorian corseting, which is why she looks so much less restricted. Her tied girdle is looser, freer and bulky, unlike Paget's highly controlled and shaped one. The looseness of her garment and her flexible pose suggest a sensuality that Paget's portraits do not.
** A skirt must be under the netting, but which fabric it is, where it stops and starts at the waist and how long it is are not clear.
** What we can see at the bottom of the skirt does not make sense: some elements appear to be disconnected or hanging unsupported in the air, and both portraits show fabric that does not seem to be a part of the other structures nearby.
** A long dark piece of fabric in the front, which is probably part of the skirt, is behind the heavy jeweled, vertically striped ornament that contributes to the stripe theme (discussed below as a striped element).
** In the album portrait her train is pulled to the front around her left side and in the portrait from the Lafayette Negative Archive around her right side, covering her feet and further confusing any attempts to understand the bottom of the skirt.
* A striped motif dominates this costume. The stripes in the headdress lappets are repeated in the girdle, in the netted skirt, in the ornament down the front and in other elements that reinforce this theme more indirectly.
** The many striped elements are not antithetical to Egyptian design, although in Egyptian images the stripes are provided by folds and pleats.
** The ornament that hangs down the center front of the skirt dominates the stripe theme. This rigid metal framework is made of vertical and horizontal stripes of metal bars and stones. The vertical stripes are strands of what looks like large diamonds or other clear stones. The horizontal stripes are made mostly of dark stones, some of which are quite large. The last set of strands is not attached at the bottom, so they appear to dangle freely. The framework is rigid both vertically and horizontally, keeping the many large jewels in place.
* ''The Gentlewoman'' says that Minnie's &quot;jewels were quite the most magnificent in the room.&quot;&lt;ref name=&quot;:72&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 32, Col. 2c}} Between the jewelry and the trim on her costume, she is virtually covered in precious stones.
** She is heavily accessorized, carrying a fan and wearing a headdress, train and a lot of jewelry, likely disassembled from other jewelry. The attendants and the long-handled fan they carried were also accessories, actually.
*** The jewelry includes the massive and ornate brooch-like or pendant pieces at the center front as well as bracelets, brooches, rings and possibly 9 necklaces.
*** Most of the stones are faceted, but the many cabochons in a diamond setting also make a repeated motif.
*** Arm bands are connected to the bracelets on her wrists by chains long enough for her to straighten her arms. Coin-like discs are attached to the armbands, bracelets and chains.
*** Some of the jewelry even looks stereotypically Egyptian, like the snake head dangling on Paget's right shoulder and the snake encircling her left upper arm.
** Her jewels, some of them strikingly large dark stones, appear both as accessories and as trim on the costume itself. Functioning both as accessories and trim, the ostentatious decorations on the center front of her costume would have been so expensive, they can only be an indicator of wealth and status. The complex netted overskirt that partially covers a solid underskirt could be analyzed as costume, accessory or trim.
*** The trim on this elaborate costume includes the shoulder straps, the striped ibis wings at the top center of the bodice, the &quot;scales&quot; on her bodice, the jeweled stripes on her headdress, the embroidered ibis wings on the train and the disconnected jewelled strands at the bottom of the skirt.
** Her hair is down and shows behind the chains behind and under her right elbow.

==== The Historical Cleopatra ====
Cleopatra lived from 70/69 B.C.E. to 10 or 12 August 30 B.C.E., the last of the Hellenistic pharaohs.&lt;ref&gt;{{Cite journal|date=2025-08-04|title=Cleopatra|url=https://en.wikipedia.org/w/index.php?title=Cleopatra&amp;oldid=1304135144|journal=Wikipedia|language=en}}&lt;/ref&gt; But nonscholarly late 19th-century Britons, Europeans and Americans would have known her less as a historical figure than a cultural one, by her presence in the arts and in popular culture. Sarah Bernhardt's performance as Cleopatra was very important, and the general discussion of the character and the popular-culture figure appears on [[Social Victorians/People/Sarah Bernhardt#Cleopatra|Bernhardt's page]]. 

=== Col. Arthur Paget ===
[[File:Sir-Arthur-Henry-Fitzroy-Paget-as-Edward-the-Black-Prince.jpg|thumb|left|alt=Black-and-white photograph of a standing man richly dressed in armor, with a sword, a cape and a helmet|Sir Arthur Henry Fitzroy Paget as Edward, the Black Prince. ©National Portrait Gallery, London.]][[File:TombaPrincepNegre.JPG|thumb|alt=Closeup of the effigy on the tomb in Canterbury Cathedral showing his armor, helmet and gloves|Effigy of Edward, the Black Prince, Canterbury Cathedral]][[File:Edward, the Black Prince, in Canterbury Cathedral 02.JPG|thumb|alt=Shield of Prince Edward on wall in Canterbury Cathedral|Coat of Arms of Edward, the Black Prince, showing fleurs de lys and lions]]
Arthur Henry Fitzroy Paget, Col. Arthur Paget, also attended, as Prince Edward of Woodstock, the &quot;Black Prince.&quot; He was married to Minnie Paget.

John Thomson's portrait (left) of &quot;Sir Arthur Henry Fitzroy Paget as Edward, the Black Prince&quot; in costume is photogravure #146 in [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|the album]] presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot; /&gt; The printing on the portrait says, &quot;Colonel Arthur Paget as Edward the Black Prince.&quot;&lt;ref&gt;&quot;Colonel Arthur Paget as Edward the Black Prince.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158509/Sir-Arthur-Henry-Fitzroy-Paget-as-Edward-the-Black-Prince.&lt;/ref&gt;

==== Newspaper Reports ====
* He was dressed as &quot;Edward the Black Prince, in a chain mail, with black velvet coat embroidered in gold, and fur coat worked with lions and fleur-de-lis in gold; black helmet and Prince of Wales's plume.&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}}

*&quot;Colonel Arthur Paget assumed the character of Edward the Black Prince in a chain-mail, with black velvet coat embroidered in gold.&quot;&lt;ref name=&quot;:5&quot; /&gt;{{rp|p. 3, Col. 3b}}

==== Commentary on His Costume ====

* Paget's costume looks appropriately 14th century, although of course contemporary methods would have been used to construct it. Also his mustache is very Victorian, not 14th century.
* The ''Morning Post'' says he is wearing a coat &quot;embroidered in gold&quot; and a coat &quot;worked with lions and fleur-de-lis,&quot; but he is not wearing a coat.&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 3, Col. 3b}} His tabard has the lions and fleurs de lys, and his cloak is edged with embroidery. He appears to be wearing velvet rather than fur. He appears to be wearing a hauberk (chain mail shirt that reaches to mid thigh) under the tabard. The hauberk has pointed [[Social Victorians/Terminology#Dags|dags]], and the tabard's dags are rounded.
* The chain mail looks very realistic except for its sheer quantity. Paget is wearing a mail coif (hood) under his helmet, a hauberk that also covers his arms, mitons on his hands and chausses on his legs. Chain mail covers even his boots. The mitons, chausses and boot covers are less common in portraits and preserved mail suits than hauberks and coifs.
* The effigy of Edward, the Black Prince in Canterbury Cathedral shows him clad in armor articulated at the knees and elbows rather than in chain mail covering his legs and arms.
* Paget's costume looks uncomfortable. The chain mail and helmet would have been heavy. Surely he would have taken off at least the cloak, helmet, hood and gloves after the procession.
* The scabbard is on the sword he is holding rather than attached to the sword belt around his waist.

==== Edward, the Black Prince ====
Edward, the Black Prince (15 June 1330 – 8 June 1376), son of King Edward III, was a successful military leader.&lt;ref&gt;{{Cite journal|date=2025-09-30|title=Edward the Black Prince|url=https://en.wikipedia.org/w/index.php?title=Edward_the_Black_Prince&amp;oldid=1314257772|journal=Wikipedia|language=en}}&lt;/ref&gt; If he had not died before his father, he would have been king of England. His tomb is in Canterbury Cathedral with his surcoat, helmet, shield, and gauntlets. Along with his coat of arms, a closeup of his effigy in Canterbury Cathedral (above, right) shows the design on the surcoat or tabard Arthur Paget is wearing.

=== Gerald Paget ===
Gerald Paget was dressed as Marc Antony in the Oriental procession (both the ''Morning Post'' and the ''Times'' call him ''Gerald Paget Paget'').&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt; No obvious candidate for Gerald Paget Paget can be found except for Gerald Cecil Stewart Paget (15 October 1854 – 25 October 1913), Sir Arthur Paget's brother, who seems quite likely, in part because he came as Antony to Minnie Paget's Cleopatra. The ''Gentlewoman'', which calls him Mr. Gerald Paget, says they walked together in the procession.&lt;ref name=&quot;:7&quot; /&gt;{{rp|p. 32, Col. 2c}}

== Demographics ==
*Nationality: Minnie (Mary) Stevens Paget was American, but Arthur Henry Fitzroy Paget was British.

== Family ==

* General Lord Alfred Henry Paget (26 June 1816 – 24 August 1888)&lt;ref&gt;&quot;General Lord Alfred Henry Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p612.htm#i6117|title=Person Page|website=www.thepeerage.com|access-date=2021-11-23}} https://www.thepeerage.com/p612.htm#i6117.&lt;/ref&gt;
* Cecilia Wyndham (baptised 1 November 1829 – 3 May 1914)&lt;ref&gt;&quot;Cecilia Wyndham.&quot; {{Cite web|url=https://www.thepeerage.com/p4699.htm#i46984|title=Person Page|website=www.thepeerage.com|access-date=2021-11-23}} https://www.thepeerage.com/p4699.htm#i46984.&lt;/ref&gt;
*# Victoria Alexandrina Paget (1848 – 2 February 1859)
*# Hon. Evelyn Cecilia Paget (c. 1850 – 17 May 1904)
*# General Rt. Hon. '''Sir Arthur Henry Fitzroy Paget''' (1 March 1851 – 8 December 1928)
*# Admiral Rt. Hon. Sir Alfred Wyndham Paget (26 March 1852 – 17 June 1918)
*# Major George Thomas Cavendish Paget (24 May 1853 – 28 January 1939)
*# Captain '''Gerald Cecil Stewart Paget''' (15 October 1854 – 25 October 1913)
*# Violet Mary Paget (1856 – 13 June 1908)
*# Lt. Sydney Augustus Paget (19 April 1857 – 16 September 1916)
*# Amy Olivia Paget (3 June 1858 – 14 February 1948)
*# Alberta Victoria Paget (1860 – 28 July 1945)
*# '''Almeric Hugh Paget, 1st and last Baron Queenborough''' (14 March 1861 – 22 September 1949)
*# Alice Maud Paget (1863 – 24 December 1925)
*# Alexandra Harriet Paget (1865 – 19 October 1944)
*# Guinevere Eva Paget (1869 – 26 February 1894)


*Arthur Henry Fitzroy Paget (1 March 1851 – 8 December 1928)&lt;ref name=&quot;:2&quot; /&gt;
*Minnie (Mary) Stevens (1853 – May 1919)&lt;ref name=&quot;:3&quot; /&gt;
#Louise Margaret Leila Wemyss Paget ( – 24 September 1958)
#Albert Edward Sydney Louis Paget (23 May 1879 – 2 August 1917)
#Arthur Wyndham Louis Paget (6 March 1888 – 28 February 1966)
#Reginald Scudamore George Paget (6 March 1888 – 11 June 1931)


* Captain Gerald Cecil Stewart Paget (15 October 1854 – 25 October 1913)&lt;ref&gt;&quot;Captain Gerald Cecil Stewart Paget.&quot; {{Cite web|url=https://www.thepeerage.com/p4697.htm#i46970|title=Person Page|website=www.thepeerage.com|access-date=2021-12-01}} https://www.thepeerage.com/p4697.htm#i46970.&lt;/ref&gt;
* Lucy Annie Emily Gardner ( – 15 April 1927)&lt;ref name=&quot;:6&quot; /&gt;
*# Dorothy Cecilia Paget (30 November 1878 – 10 February 1936)
*# Lettice Mina Paget (25 July 1880 – 6 December 1969)

=== Stevens Family ===

* Paran Stevens (11 September 1802 – 25 April 1872)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/93965613/paran-stevens|title=Paran Stevens (1802-1872) - Find a Grave Memorial|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/93965613/paran-stevens.&lt;/ref&gt;
* Eliza Jewett (1 April 1801 – 4 March 1850)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/145086867/eliza-stevens|title=Eliza Jewett Stevens (1801-1850) - Find a Grave...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/145086867/eliza-stevens.&lt;/ref&gt;
*# Ellen Stevens Melcher (September 1826 – 11 September 1908)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/142694577/ellen-melcher|title=Ellen Stevens Melcher (1826-1908) - Find a Grave...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/142694577/ellen-melcher.&lt;/ref&gt;
* Marietta Reed (1827 – 3 April 1895)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/93965727/marietta-stevens|title=Marietta Reed Stevens (1827-1895) - Find a Grave...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/93965727/marietta-stevens.&lt;/ref&gt;
*# '''Minnie (Mary) Fiske Stevens''' (13 August 1853 – 20 May 1919)&lt;ref&gt;{{Cite web|url=https://www.findagrave.com/memorial/74543676/mary-fiske-paget|title=Mary Fiske “Minnie” Stevens Paget (1853-1919) -...|website=www.findagrave.com|language=en|access-date=2023-09-17}} https://www.findagrave.com/memorial/74543676/mary-fiske-paget.&lt;/ref&gt;

== Questions and Notes ==
#Arthur Fitzroy Paget was a colonel in 1897, a general in 1902.
#Mrs. Arthur Paget is #90 and Mr. Arthur Paget is #91 [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|in the list of people present]] at the [[Social Victorians/1897 Fancy Dress Ball |Duchess of Devonshire's fancy-dress ball]]. Gerald Paget is #237.

== Bibliography ==

# Fitzroy, Sir Almeric. ''Memoirs''. 2 vols. 1925.

== Footnotes ==
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    <title>Social Victorians/People/Sarah Spencer-Churchill Wilson</title>
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      <text bytes="23906" sha1="roh7e51jpqgegalq02q2entgtxt5ddv" xml:space="preserve">== Overview ==
== Acquaintances, Friends and Enemies ==
== Organizations ==

=== Lady Sarah Wilson ===
*&quot;[[Social Victorians/People/Working in Publishing#Journalists|aristocratic lady journalist]]&quot;
*Lady Sarah Wilson, journalist for the ''Daily Mail''&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2020-07-06|title=Sarah Wilson (war correspondent)|url=https://en.wikipedia.org/w/index.php?title=Sarah_Wilson_(war_correspondent)&amp;oldid=966295858|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Gordon Wilson ===
*Gordon Wilson, Royal Horse Guards
*Gordon Wilson, Robert Baden-Powell's aide de camp at Mafeking

=== Wilfred Wilson ===

* 5th Battalion Imperial Yeomanry

== Timeline ==
'''1861''', Sir Samuel Wilson and Jeanne Campbell married.&lt;ref name=&quot;:2&quot;&gt;&quot;Sir Samuel Wilson.&quot; {{Cite book|url=https://books.google.com/books?id=KDw6AQAAMAAJ|title=Armorial Families: A Complete Peerage, Baronetage, and Knightage, and a Directory of Some Gentlemen of Coat-armour, and Being the First Attempt to Show which Arms in Use at the Moment are Borne by Legal Authority|last=Fox-Davies|first=Arthur Charles|date=1895|publisher=Jack|language=en}} 1047, Col. 1a.&lt;/ref&gt;

'''1891 November 21''', Sarah Isabella Augusta Spencer-Churchill and Gordon Chesney Wilson married.&lt;ref&gt;&quot;Lady Sarah Isabella Augusta Spencer-Churchill.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106326|title=Person Page|website=www.thepeerage.com|access-date=2020-10-20}}&lt;/ref&gt;

'''1892 June 11''', Adeline Constance Wilson and Right Hon. the Earl of Huntingdon married.&lt;ref name=&quot;:2&quot; /&gt;

'''1897 July 2, Friday''', Lady Sarah Wilson and Captain Gordon Wilson attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did Mr. Wilfred Wilson, Mr. Clarence Wilson, and Mr. Herbert Wilson.
[[File:Madame de Pompadour.jpg|alt=Old painting of a woman in a very ornate dress with an open book|thumb|Madame de Pompadour, 1756, ]]

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
=== Lady Sarah Wilson ===
[[File:Lady-Sarah-Isabella-Augusta-Wilson-ne-Spencer-Churchill-as-Madame-de-Pompadour.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a dog|Lady Sarah Wilson as Madame de Pompadour. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lady Sarah Wilson went as Madame de Pompadour.&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and https://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7c}} 

John Thomson's portrait (left) of &quot;Lady Sarah Isabella Augusta Wilson (née Spencer-Churchill) as Madame de Pompadour&quot; in costume is photogravure #157 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|album]] presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Sarah Wilson as Madame de Pompadour.&quot;&lt;ref&gt;&quot;Lady Sarah Wilson as Madame de Pompadour.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158520/Lady-Sarah-Isabella-Augusta-Wilson-ne-Spencer-Churchill-as-Madame-de-Pompadour.&lt;/ref&gt; 

If Lady Sarah Wilson's dress is indeed blue, as the descriptions say, then Thomson's portrait is an excellent example of how difficult it can be to guess the colors of things in black-and-white photographs. Although the album (and the National Portrait Gallery, London) credit Thomson for the photograph, the portrait of Lady Sarah from the album looks more like a painting than a photograph. Perhaps it was retouched to make it look less photographic and more painterly. 

Surprisingly, two portraits of Lady Sarah appear in the Lafayette Archive, suggesting that she also had her photograph taken by the Lafayette firm, perhaps at the ball itself. The Lafayette Archive lists 2 photographs but provides only one: 

* http://lafayette.org.uk/wil1366.html

This image is a higher resolution and more clear, and it is not retouched to appear more like a painted portrait. Not all particulars of her costume are identical in the Lafayette and Thomson portraits. 

Another image of Lady Sarah Wilson in costume appeared in the ''Queen'' (bottom middle of the page, the numeral 17 below the line drawing, seated, facing slightly to her right, the drawing shows a dress similar to her costume in her photograph, bows and ruffles emphasized; the drawing apparently signed by “Rook”).&lt;ref name=&quot;:8&quot;&gt;“Dresses Worn at the Duchess of Devonshire’s Fancy Ball on July 2.” The ''Queen'', The Lady’s Newspaper 10 July 1897, Saturday: 52 [of 98 BNA; p. 78 on printed page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/174/0052.&lt;/ref&gt;{{rp|Col. 2b–c}} 

François Boucher's 1756 portrait of Madame de Pompadour (above right) shows Jeanne Antoinette Poisson, Madame de Pompadour at about 35 years old.&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|date=2023-12-13|title=Madame de Pompadour|url=https://en.wikipedia.org/w/index.php?title=Madame_de_Pompadour&amp;oldid=1189755757|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Madame_de_Pompadour.&lt;/ref&gt; Lady Sarah Wilson was nearly 32 years old at the time of the ball. (The color of the dress in this image may not be true to the painting; a different copy shows it looking bluer.&lt;ref&gt;{{Cite journal|date=2023-12-13|title=Madame de Pompadour|url=https://en.wikipedia.org/w/index.php?title=Madame_de_Pompadour&amp;oldid=1189755757|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Madame_de_Pompadour#/media/File:Madame_de_Pompadour.jpg&lt;/ref&gt;) 
[[File:François Boucher - Portrait of Marquise de Pompadour - WGA02909.jpg|thumb|Madame de Pompadour, Boucher, 1759, with Friendship's consolation of Love behind her]]
Another Boucher portrait of Madame de Pompadour (right), painted in 1759 when she was 38,&lt;ref name=&quot;:7&quot; /&gt; shows her in a very similar dress, though pink and yellow rather than blue or blue-green. We can see how the skirt falls when she is standing. 

==== Madame de Pompadour ====
Politically active, Madame de Pompadour was Louis XV's official chief mistress until 1751 and lady in waiting to the Queen, Polish Marie Leszczyńska.&lt;ref name=&quot;:7&quot; /&gt; She was leader of fashionable society until Louis XV's death and Marie Antoinette's rise displaced her.

==== Newspaper Accounts ====
Most of the descriptions of Lady Sarah Wilson's costume were published in fashion rather than news perioodicals, unlike the descriptions of politically important people.
* &quot;(Mme. de Pompadour), blue and magenta, silk, lace, and pink roses; bunch of wild hyacinths, yellow daisies, and pink roses on left shoulder.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 40, Col. 2b}}
* The ''Queen'' has 2 descriptions, this one which is included in the descriptions of the &quot;general company&quot; and the one below, highlighting the dressmaker, Mrs Mason:&lt;blockquote&gt;Lady Sarah Wilson wore a Pompadour costume of rich china-blue satin, the quaint bodice with deep point in front, fastened with old-fashioned bows of vieux-rose silk, graduating in size to the waist; the tight satin sleeves had deep frills of silk, pinked at the edged at the elbow with an inner frill of lace; the dress was trimmed with white blonde lace and pink Banksia roses; the skirt was of blue satin, with very full paniers, and flounced with two frills, edged with blonde lace and pink button roses.&lt;ref&gt;“Dresses Worn at the Duchess of Devonshire’s Fancy Ball on July 2.” The ''Queen'', The Lady’s Newspaper 10 July 1897, Saturday: 50 [of 98 BNA; p. 76 on printed page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/171/0050.&lt;/ref&gt;{{rp|Col. 2b}}&lt;/blockquote&gt;
* The description accompanying the line drawing in the ''Queen'' says the original was owned by Baron Ferdinand de Rothschild, which means that Boucher's blue-dress portrait (above right) is the original:&lt;blockquote&gt;Made by Mrs Mason, 4, New Burlington Street, W. …   No. 17. L&lt;small&gt;ADY&lt;/small&gt; S&lt;small&gt;ARAH&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, Madame de Pompadour (copied from the picture of “La Pompadour” of Baron Ferdinand de Rothschild). — Rich / blue satin, with ruchings of satin and white blonde lace, with wreath of roses; Alençon lace ruffles; headdress, small wreath of roses, with high aigrette.&lt;ref name=&quot;:8&quot; /&gt;{{rp|Col. 2–3c}}&lt;/blockquote&gt;
==== Commentary on Lady Sarah's Costume ====
These descriptions are based on the Thomson portrait published in the commemorative album (above left).
* Lady Sarah is holding her skirt in her left hand oddly, making the layers of the skirt confusing but suggesting that the overskirt has no trim other than what is at the opening.
* The dresses in the Boucher portraits are very similar to each other, but the blue 1856 one is the original for Lady Sarah's dress.&lt;ref&gt;{{Cite web|url=http://lafayette.org.uk/wil1366.html|title=Lady Sarah Wilson at the Devonshire House Ball 1897, by Lafayette|website=lafayette.org.uk|access-date=2026-05-13}}&lt;/ref&gt;
* The skirts in the Boucher portraits are voluminous, unlike the skirt Lady Sarah is wearing, which may be influenced by 1890s style, whose close-fitted skirts had a smooth, bell-shaped flare.&lt;ref&gt;Matthews, Mimi. A Victorian Lady's Guide to Fashion and Beauty. Pen &amp; Sword History, 2018.&lt;/ref&gt;{{rp|73}} She may be wearing paniers (or a bum-roll), but like the skirt they are more modest than what Madame de Pompadour is wearing in the Boucher portraits. Or perhaps the modesty in Lady Sarah's costume means that it was less expensive? Or that she, appropriately, did not want to compete with the opulence of the costume of Daisy, Countess Warwick as Marie Antoinette?
* According to the description, the bows on the bodice — or eschelles — are &quot;graduating in size to the waist,&quot; but in fact they diminish in size.
* In some respects, this costume is an 18th-century design: the graduated bows in the bodice, the multiple layers of ruffled lace in the sleeves, the overskirt and petticoat construction, the v-point below the waist of the bodice, the double-ruffle and flower trim on the skirt and bodice and the piled-up powdered hair with ringlets. The symmetry of the dress is consistent with 18th-century design. The design has 18th-century elements, but the line of the skirt is not 18th or 19th century.
* According to the ''Queen'', the roses on Lady Sarah's dress were Banksia roses, ''Rosa banksiae'', which have more, frillier petals than the long-stemmed roses we're accustomed to seeing, and they grow in clusters on short stems on longer trailing stems.
* As in the Pompadour portraits, Lady Sarah is accompanied by a small dog.
* The large cluster of flowers on her left shoulder breaks the symmetry of the design of her costume.
[[File:Gordon-Chesney-Wilson-as-a-Captain-in-the-Blues-1680.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume|Gordon Chesney Wilson as a Captain in the Blues, 1680. ©National Portrait Gallery, London.]]

=== Captain Gordon Wilson ===
Most newspapers say Captain Gordon Wilson was in costume as a member of the Royal Horse Guard of John Churchill, 1st Duke of Marlborough (1650–1722&lt;ref&gt;{{Cite journal|date=2023-12-03|title=John Churchill, 1st Duke of Marlborough|url=https://en.wikipedia.org/w/index.php?title=John_Churchill,_1st_Duke_of_Marlborough&amp;oldid=1188192102|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/John_Churchill,_1st_Duke_of_Marlborough.&lt;/ref&gt;). According to the typographical visualization of the quadrilles and processions in the ''Morning Post'', however, Captain Gordon Wilson was one of the Mousquetaires et Militaires de l'Epoque in the Louis XV and Louis XVI Quadrille, along with Sir Samuel Scott.&lt;ref name=&quot;:3&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|7, Col. 6b}} But only one newspaper says he was a Mousquetaire.

Lafayette's portrait of &quot;Gordon Chesney Wilson as a Captain in the Blues, 1680&quot; in costume is photogravure #158 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:1&quot; /&gt; The printing on the portrait says, &quot;Captain Gordon Wilson as a Captain in the Blues temp 1680.&quot;&lt;ref&gt;&quot;Captain Gordon Wilson as a Captain in the Blues.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158521/Gordon-Chesney-Wilson-as-a-Captain-in-the-Blues-1680.&lt;/ref&gt;

The Blues were the Royal Regiment of Horse Guards, part of the [[Social Victorians/Terminology#Household Cavalry|Household Cavalry]]: the coat was blue, with red facings, collar and plumes.&lt;ref&gt;{{Cite journal|date=2021-11-11|title=Royal Horse Guards|url=https://en.wikipedia.org/w/index.php?title=Royal_Horse_Guards&amp;oldid=1054735721|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Royal_Horse_Guards.&lt;/ref&gt;

==== Newspaper Descriptions of His Costume ====
*He wore a &quot;Costume of his own regiment at the time of the Duke of Marlborough, blue with red facings, embroidered gold crimson sash, and embroidered baldric, large velvet hat and plumes.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|p. 8, Col. 1c}}
*&quot;Sir Samuel Scott and Captain Gordon Wilson [wore] uniforms of the R.H.G. [Royal Horse Guards] in the great Duke of Marlborough's time.&quot;&lt;ref&gt;“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.&lt;/ref&gt;{{rp|42, Col. 2b}}
*&quot;Captain Gordon Wilson and Sir Samuel Scott (costume of their own regiments at the time of the Duke of Marlborough), blue with red facings; velvet hat and plumes.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 36, Col. 3b}}

==== Commentary on His Costume ====

* Gordon Chesney Wilson seems to have been a member of the [[Social Victorians/Terminology#Royal Horse Guards|Royal Horse Guards]] and wore a 17th-century uniform to the ball.
* This is not the uniform of a captain dressed for battle. Wilson is in court dress. The shoes, for example, are court shoes with a high tongue, a large buckle and bow in the buckle, and possibly red heels. His jabot (neck treatment) is appropriate for court dress of c. 1680, as are his curly wig and the bows on the knee bands of his breeches and at the shoulders. Wilson's shirt has full sleeves that are gathered into lacy ruffles at the wrist and are pulled out over the hands from the cuffs of the jacket.
* Wilson's costume has some Cavalier elements, appropriately, but it is less ornate than non-military outfits would have been.
* The embroidered or appliquéd trim is the same on the cuffs, the front of the jacket and the baldric — a distinctive curled feather shape. The wide decorated cuffs on the jacket were fashionable at the end of the 17th century.
=== Wilfred Wilson ===
Wilfred Wilson was among the Suite of Men in the &quot;Oriental&quot; procession.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; The ''Gentlewoman'' says, &quot;Messrs [[Social Victorians/People/Halifax|Gordon Wood]] and Wilfred Wilson were attendants on [&lt;nowiki/&gt;[[Social Victorians/People/Keppel|George Keppel]]'s] King Solomon,&quot; wearing &quot;green silk tunics elaborately embroidered in gold and studs, with cloaks embroidered and lined with white; jewelled headdresses, swords.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 34, Col. 3a}} No photograph of him in costume can be found at this time.[[File:Jean Fouquet- Portrait of the Ferrara Court Jester Gonella.JPG|alt=Old painting of the face and upper body of an Italian court jester|thumb|Ferrara Court Jester Gonella, in the style of Albrecht Dürer|left]]

=== Clarence Wilson ===
Mr. Clarence Wilson, likely Chesney Clarence Wilson?, was dressed as Buffone in the Venetians procession.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt; Buffone was a stock comic character, the clown. A c. 1450 portrait of professional jester Gonella in the style of Albrecht Dürer (right) shows one such Buffone.

* &quot;Mr. Clarence Wilson (jester), in satin, with gold thread embroidery.&quot;&lt;ref name=&quot;:6&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. Print p. 50, Col. 3a.&lt;/ref&gt;{{rp|p. 34, Col. 1b}}
[[File:Attributed to Odoardo Fialetti (1573-1638) - Doge Antonio Priuli - RCIN 407153 - Royal Collection.jpg|alt=Old painting of a man dressed in a white tunic, a red and gold cape and hat, and gloves and a beard|thumb|Antonio Priuli, Doge of Venice 1618–1623]]
=== Herbert Wilson ===
Mr. Herbert Wilson was dressed as Antonio Priali&lt;ref name=&quot;:3&quot; /&gt; (misspelled as Briali&lt;ref name=&quot;:4&quot; /&gt;&lt;ref&gt;“Ball at Devonshire House.” Evening ''Mail'' 05 July 1897 Monday: 8 [of 8], Col. 1a–4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003187/18970705/070/0008.&lt;/ref&gt;{{rp|p. 8, Col. 1c}}) in the Venetians procession. A c. 1600–1625 portrait of Priuli (left) shows him richly dressed.

* &quot;Mr. Herbert Wilson (Venetian noble), vieux rose brocaded velvet.&quot;&lt;ref name=&quot;:6&quot; /&gt;{{rp|p. 34, Col. 1b}}

=== Wilsons Who Attended by Family ===

==== Lady Sarah and Captain Gordon Wilson Family ====

* Lady Sarah Wilson and Captain Gordon Wilson
* Mr. Wilfred Wilson
* Mr. Clarence Wilson
* Mr. Herbert Wilson

==== [[Social Victorians/People/Arthur Stanley Wilson|Arthur Stanley and Mary Wilson Family]] ====

* Arthur and Mary Wilson
* Clive Wilson
* Tottie (Susannah West) Wilson Menzies and Jack Graham Menzies
* [[Social Victorians/People/Muriel Wilson|Muriel Wilson]]
* Mr. and Mrs. Charles Henry Wilson
* Enid Wilson

==== Unknown Family ====

* Mr. T.W. Wilson

== Demographics ==
*Nationality: she, English&lt;ref name=&quot;:0&quot; /&gt;; he, Australian
*Samuel Wilson, born in Ireland, his wife and many of children born in Australia&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2020-03-15|title=Samuel Wilson (Portsmouth MP)|url=https://en.wikipedia.org/w/index.php?title=Samuel_Wilson_(Portsmouth_MP)&amp;oldid=945720739|journal=Wikipedia|language=en}}&lt;/ref&gt;

=== Residences ===

==== Sir Samuel Wilson ====

* After returning from Australia
* 9 Grosvenor Square, London (March 1895 – 11 June 1895)&lt;ref name=&quot;:2&quot; /&gt;
* Hughenden Manor, High Wycombe, Bucks (1881– September 1893?)&lt;ref name=&quot;:2&quot; /&gt;

== Family ==
=== Gordon Chesney Wilson's Family ===

* Sir Samuel Wilson (7&lt;ref name=&quot;:2&quot; /&gt; or 17&lt;ref name=&quot;:9&quot;&gt;Ancestry.com. ''UK and Ireland, Find a Grave® Index, 1300s-Current'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2012.&lt;/ref&gt; February 1832 – 11 June 1895)&lt;ref name=&quot;:5&quot; /&gt;
* Jeanne Campbell, Lady Wilson (8 May 1841 – 8 February 1925)&lt;ref name=&quot;:9&quot; /&gt;
*# '''Gordon Chesney Wilson''' (1 August 1865 – 6 November 1914)
*# Mary Wilson (c. 1870 –&lt;ref name=&quot;:10&quot;&gt;''Census Returns of England and Wales, 1901''. Kew, Surrey, England: The National Archives, 1901. Class: ''RG13''; Piece: ''82''; Folio: ''199''; Page: ''49''.&lt;/ref&gt; )
*# '''Wilfred Wilson''' (3 March 1872 – February 1901&lt;ref&gt;&quot;Man and Matters.&quot; ''Globe'' 26 February 1901 Tuesday: 3 [of 10], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/19010226/030/0003.&lt;/ref&gt;)
*# '''Clarence Chesney Wilson''' (2 March 1873 – )
*# Bertie (Herbert Hayden) Wilson (4 February 1875 – )
*# Adeline Constance Wilson Lloyd (c. 1867&lt;ref&gt;''Census Returns of England and Wales, 1901''. Kew, Surrey, England: The National Archives, 1901. Class: ''RG13''; Piece: ''82''; Folio: ''198''; Page: ''48''.&lt;/ref&gt;– 24 October 1933&lt;ref&gt;Principal Probate Registry; London, England; ''Calendar of the Grants of Probate and Letters of Administration made in the Probate Registries of the High Court of Justice in England''. Ancestry.com. ''England &amp; Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1995'' [database on-line]. Lehi, UT, USA: Ancestry.com Operations, Inc., 2010.&lt;/ref&gt;)
*# Maud Margaret Wilson (1870&lt;ref name=&quot;:11&quot;&gt;The National Archives of the UK (TNA); Kew, Surrey, England; ''Census Returns of England and Wales, 1891''; Class: ''RG12''; Piece: ''68''; Folio: ''21''; Page: ''38''; GSU roll: ''6095178''.&lt;/ref&gt;– ) [Maud, Countess Huntington?&lt;ref name=&quot;:10&quot; /&gt;]
*# Florence Mabel Wilson ()
*# Herbert H. Wilson (1878&lt;ref name=&quot;:11&quot; /&gt;–) [see Bertie, above]


*Sarah Isabella Augusta [[Social Victorians/People/Marlborough | Spencer-Churchill]] Wilson (4 July 1865 – 22 October 1929)
*Gordon Chesney Wilson (1 August 1865 – 6 November 1914)&lt;ref&gt;&quot;Lt.-Col. Gordon Chesney Wilson.&quot; {{Cite web|url=https://www.thepeerage.com/p10633.htm#i106327|title=Person Page|website=www.thepeerage.com|access-date=2020-10-20}}&lt;/ref&gt;
#Randolph Gordon Wilson (1893–1956)&lt;ref name=&quot;:0&quot; /&gt;

=== Relations ===

* Sarah Isabella Augusta Spencer-Churchill's brothers were [[Social Victorians/People/Churchill|Lord Randolph Churchill]] and Sunny (Charles Richard John) Spencer-Churchill, [[Social Victorians/People/Marlborough|9th Duke of Marlborough]] (9 November 1892 – 30 June 1934).

== Also Known As ==
*Family name: Wilson
*Sarah Isabella Augusta [[Social Victorians/People/Marlborough | Spencer-Churchill]]
*Captain Gordon Wilson, M.V.O.
*Lady Sarah Wilson
*The family of [[Social Victorians/People/Arthur Stanley Wilson|Arthur Stanley Wilson]]

== Questions and Notes ==
#Lady Sarah Wilson is the 11th child and 6th daughter of John Winston Spencer-Churchill, 7th [[Social Victorians/People/Marlborough | Duke of Marlborough]] and Frances Anne Emily Vane Spencer-Churchill, [[Social Victorians/People/Marlborough | Duchess of Marlborough]].
#Lady Sarah Wilson is one of the &quot;aristocratic lady journalists&quot; and was at Mafeking with her husband, Capt. Gordon Wilson.
#Gordon Chesney Wilson died in at the first battle of Ypres, 6ths November 1914.
#For the Samuel Wilson family, any Miss Wilson after 1892 has to have been Florence Mabel Wilson.
#Three somewhat difficult-to-identify men were among the Suite of Men in the &quot;Oriental&quot; procession: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]], Wilfred Wilson, and [[Social Victorians/People/Bourke|Hon. Algernon Bourke]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts; the Hon. Algernon Bourke is not difficult to identify at all; Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#There is a problem with Herbert Hayden Wilson and Herbert H. Wilson's birth dates.
#Captain Gordon Wilson is #96 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] the[[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House; Lady Sarah Wilson is #392; Wilfred Wilson is #232; Mr. Clarence Wilson is #300; Mr. Herbert Wilson is #307.

== Footnotes ==
{{reflist}}</text>
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  </page>
  <page>
    <title>Irish Language/Similar Words in Irish and English</title>
    <ns>0</ns>
    <id>268762</id>
    <revision>
      <id>2810724</id>
      <parentid>2716778</parentid>
      <timestamp>2026-05-21T03:33:51Z</timestamp>
      <contributor>
        <username>ABehrens</username>
        <id>434601</id>
      </contributor>
      <comment>Add a near-cognate for ceathair.</comment>
      <origin>2810724</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3271" sha1="fhoeik93yxypn6pezu93bx85ovbry7c" xml:space="preserve">The English and Irish words in this list generally agree sufficiently in both spelling and pronunciation between the languages to be immediately recognisable, and also have the same meaning in both languages.  They therefore can be of help, especially for the novices.

The words in the word pairs in this list are similar, since in 'fairly recent time' one of them was borrowed from the other, or both were borrowed from a third language.  'Fairly recent time' here in general stands for 'At most a few hundred years back'.  There are also word pairs which are related, but where the common ancestor of the words split at an earlier time.  In fact, there are many such words, since both English and Irish ultimately derive from a common [[w:Proto-Indo-European language|Proto-Indo-European language]], spoken perhaps five thousand years ago.  However, both language branches changed radically in these millennia, and most such '[[w:cognate|cognate]]s' are unrecognizable to-day, without studying a bit of philology.  For examples, English ''three'' and Irish ''trí'' are cognates, originating from the same Proto-Indo-European word, as are ''four'' and ''ceathair''.  The first pair actually might be recognized, but hardly the second one (although the resemblance to ''quarter'' is easier to see).. Many other words appear to be Irish borrowings into English, but are actually both languages borrowing from French or Latin. An interesting example is ''carry'' - apparently a direct borrowing of English ''car'', both words actually have their root in Gaulish ''carros'' (&quot;two-man chariot&quot;).

{| class=&quot;wikitable&quot;
|-
! English !! Irish
|-
| able || ábalta
|-
| abnormal || mi-normálta
|-
| banana || banana
|-
| bar || barra
|-
| bolt || bolta
|-
| brandy || branda
|-
| bulb || bulba
|-
| cable || cábla
|-
| camp || campa
|-
| can || canna
|-
|car
|carr
|-
| card || carta
|-
| clamp || clampa
|-
| cord || corda
|-
| coast || cósta
|-
| coat || cóta
|-
| crock || croca
|-
| cross || crosta
|-
| crust || crusta
|-
| cube || ciúb
|-
| drum || druma
|-
| drug || druga
|-
| elephant || eilifint
|-
| force || fórsa
|-
| form || forma
|-
| gate || geata
|-
| gross || grósa
|-
| group || grúpa
|-
| gun || gunna
|-
| hall || halla
|-
| hat || hata
|-
| lamp || lampa
|-
|lion
|leon
|-
| monkey || moncaí
|-
| muff || mufa
|-
| mug || muga
|-
| note || nóta
|-
| pack || paca
|-
| pearl || péarla
|-
| pan || panna
|-
| pin || pionna
|-
| piece || piosa
|-
| plug || pluga
|-
| plum || pluma
|-
| plate || plata
|-
| pot || pota
|-
| pump || pumpa
|-
| plug || pluga
|-
| prune || prúna
|-
| raft || rafta
|-
| ramp || rampa
|-
| rasp || raspa
|-
| robe || roba
|-
| roll || rolla
|-
| roast || rósta
|-
 
|-
| root || ruta
|-
| rope || rópa
|-
| rug || ruga
|-
| rung || runga
|-
| sample || sampla
|-
| scroll || scrolla
|-
| sloop || slúpa
|-
| slug || sluga
|-
| spoke || spóca
|-
| spot || spota
|-
| stall || stalla
|-
| stamp || stampa
|-
| strap || strapa
|-
| stump || stumpa
|-
| 
|-
| tar || tarra
|-
| theme || téama
|-
| ton || tonna
|-
| turnip || tornapa
|-
| toast || tósta
|-
| tug || tuga
|-
| volt || volta
|-
| vote || vóta
|-
| yoghurt || iógart
|-
|}

[[Category:Irish|Similar Words in Irish and English]]
[[Category:English]]</text>
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  </page>
  <page>
    <title>Template:Regular convex 4-polytopes</title>
    <ns>10</ns>
    <id>272877</id>
    <revision>
      <id>2810641</id>
      <parentid>2806184</parentid>
      <timestamp>2026-05-20T18:30:50Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>cite Coxeter on Schläfli symbol</comment>
      <origin>2810641</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="26887" sha1="jmqd0aupwz1l9ai3xmace3ysozvsoe7" xml:space="preserve">{| class=&quot;wikitable mw-collapsible {{{collapsestate|mw-collapsed}}}&quot; style=&quot;white-space:nowrap;text-align:center;&quot;
!colspan={{{columns|7}}}|{{#ifeq:{{{columns|7}}}|7|Sequence of 6 [[{{{wiki|}}}Regular 4-polytopes|regular convex 4-polytopes]]|{{#ifeq:{{{columns}}}|9|Sequence of 8 regular-faceted convex 4-polytopes|...}}}} {{#if:{{{radius|}}}|of radius {{{radius|}}}|}}
|-
!style=&quot;text-align:right;&quot;|[[{{{wiki|}}}Coxeter_group|Symmetry group]]
|[[{{{wiki|}}}Tetrahedral symmetry|A&lt;sub&gt;4&lt;/sub&gt;]]
|colspan=2|[[{{{wiki|}}}Hyperoctahedral_group|B&lt;sub&gt;4&lt;/sub&gt;]]
|[[{{{wiki|}}}F4_(mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]]
|colspan={{#ifeq:{{{columns|7}}}|7|2|{{#ifeq:{{{columns}}}|9|4|2}}}}|[[{{{wiki|}}}H4_polytope|H&lt;sub&gt;4&lt;/sub&gt;]]
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Name
|style=&quot;vertical-align:top;&quot;|[[5-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Tetrahedron|tetrahedron]]&lt;BR&gt;
5-point
|style=&quot;vertical-align:top;&quot;|[[16-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Octahedron|octahedron]]&lt;BR&gt;
8-point
{{!}}style=&quot;vertical-align:top;&quot;{{!}}[[{{{wiki|}}}8-cell|8-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Cube|cube]]&lt;BR&gt;
16-point
{{!}}style=&quot;vertical-align:top;&quot;{{!}}[[24-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Cuboctahedron|cuboctahedron]]&lt;BR&gt;
24-point
{{!}}style=&quot;vertical-align:top;&quot;{{!}}[[600-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Regular icosahedron|icosahedron]]&lt;BR&gt;
120-point
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}style=&quot;vertical-align:top;&quot;{{!}}[[{{{wiki|}}}11-cell|11-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Buckminsterfullerene|...]]&lt;BR&gt;
11-point
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}style=&quot;vertical-align:top;&quot;{{!}}[[#One fibration of 11-cells|...-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Rhombic triacontahedron|...]]&lt;BR&gt;
137-point
|}}}}
|style=&quot;vertical-align:top;&quot;|[[120-cell]]&lt;BR&gt;
Hyper-[[{{{wiki|}}}Regular dodecahedron|dodecahedron]]&lt;BR&gt;
600-point
|-
!style=&quot;text-align:right;&quot;|[[{{{wiki|}}}Schläfli symbol|Schläfli symbol]]{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}}

|{3, 3, 3}
|{3, 3, 4}
|{4, 3, 3}
|{3, 4, 3}
|{3, 3, 5}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}&lt;sub&gt;5&lt;/sub&gt;{3, 5, 3}&lt;sub&gt;5&lt;/sub&gt;
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|{5, 3, 3}
|-
!style=&quot;text-align:right;&quot;|[[{{{wiki|}}}Coxeter diagram|Coxeter mirrors]]
|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}}
|{{Coxeter–Dynkin diagram|node_1|4|node|3|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
|{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}
|-
!style=&quot;text-align:right;&quot;|Mirror dihedrals
|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|4}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|{{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|5}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|{{sfrac|𝝅|5}} {{sfrac|𝝅|3}} {{sfrac|𝝅|3}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}} {{sfrac|𝝅|2}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Graph
|[[Image:4-simplex t0.svg|120px]]
|[[Image:4-cube t3.svg|120px]]
|[[Image:4-cube t0.svg|120px]]
|[[Image:24-cell t0 F4.svg|120px]]
|[[Image:600-cell graph H4.svg|120px]]{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|[[Image:120-cell graph H4.svg|120px]]
|-
!style=&quot;text-align:right;&quot;|Vertices{{Efn|{{#if:{{{instance|}}}||The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more 4-content within the same radius.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[120-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in the ascending sequence that begins with the 5-point (5-cell) 4-polytope and ends with the 600-point (120-cell) 4-polytope.}}|name=4-polytopes ordered by size and complexity}}
|5 tetrahedral
|8 octahedral
|16 tetrahedral
|24 cubical
|120 icosahedral{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}11 hemi-dodecahedral
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|600 tetrahedral
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[120-cell#Chords|Edges]]
|10 triangular
|24 square
|32 triangular
|96 triangular
|720 pentagonal{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}55 triangular
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|1200 triangular
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Faces
|10 triangles
|32 triangles
|24 squares
|96 triangles
|1200 triangles{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}55 triangles
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}2055 golden rhombi(?)
|}}}}
|720 pentagons
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Cells
|5 {3, 3}
|16 {3, 3}
|8 {4, 3}
|24 {3, 4}
|600 {3, 3}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}11 {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3&lt;nowiki/&gt;}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}137 {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 5&lt;nowiki/&gt;}
|}}}}
|120 {5, 3}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[600-cell#Clifford parallel cell rings|Tori]]
|[[5-cell#Boerdijk–Coxeter helix|5 {3, 3}]]
|[[16-cell#Helical construction|8 {3, 3}]] x 2
|[[{{{wiki|}}}8-cell#Construction|4 {4, 3}]] x 2
|[[24-cell#Cell rings|6 {3, 4}]] x 4
|[[600-cell#Boerdijk–Coxeter helix rings|30 {3, 3}]] x 20{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}[[#Cell rings of the 11-cells|11 {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 5&lt;nowiki/&gt;}]]
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}11 [[#The 137-point ...-cell|137 {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3&lt;nowiki/&gt;}]]
|}}}}
|[[120-cell#Intertwining rings|10 {5, 3}]] x 12
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Inscribed
{{#ifeq:{{{columns|7}}}|7|
{{!}}120 in 120-cell
{{!}}675 in 120-cell
{{!}}2 16-cells
{{!}}3 8-cells
{{!}}25 24-cells
{{!}}10 600-cells
|{{#ifeq:{{{columns}}}|9|
{{!}}120 in 120-cell&lt;br&gt;96 in ...-cell
{{!}}2 5-cells&lt;br&gt;675 in 120-cell
{{!}}2 16-cells&lt;br&gt;12 5-cells
{{!}}3 8-cells&lt;br&gt;3 16-cells
{{!}}25 24-cells&lt;br&gt;75 8-cells
{{!}}5 16-cells&lt;br&gt;6 5-cells
{{!}}11 11-cells +&lt;br&gt;4 24-cells
{{!}}.. 11-cells&lt;br&gt;10 600-cells
|...}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}-
!style=&quot;vertical-align:top;text-align:right;&quot;{{!}}Pentads
{{!}}1
{{!}}2
{{!}}12
{{!}}16
{{!}}48
{{!}}6
{{!}}96
{{!}}120
{{!}}-
!style=&quot;vertical-align:top;text-align:right;&quot;{{!}}Hexads
{{!}}
{{!}}2
{{!}}12
{{!}}16
{{!}}200
{{!}}5
{{!}}480
{{!}}600
{{!}}-
!style=&quot;vertical-align:top;text-align:right;&quot;{{!}}Heptads
{{!}}
{{!}}
{{!}}
{{!}}4
{{!}}11
{{!}}4
{{!}}55
{{!}}120
|}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[{{{wiki|}}}Great circle|Great polygons]]
|
|2 [[16-cell#Coordinates|squares]] x 3{{Efn|{{#if:{{{instance|}}}||In 4 dimensional space we can construct 4 pairwise perpendicular axes and 6 pairwise perpendicular planes through a point.
Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system.
In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz).
Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''[[{{{wiki|}}}Completely orthogonal|completely orthogonal]]'' to just one of the others: the only one with which it does not share an axis.
Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.}}|name=Six orthogonal planes of the Cartesian basis}}
|4 [[120-cell#Geodesic rectangles|rectangles]] x 4
|4 [[24-cell#Great hexagons|hexagons]] x 4
|12 [[600-cell#Decagons|decagons]] x 6{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}1 [[{{{wiki|}}}Hendecagon|11-gon]] x 1
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}11 [[#Cell rings of the 11-cells|11-gons]] x 11
|}}}}
|100 [[120-cell#Compound of five 600-cells|irregular hexagons]] x 4
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|[[{{{wiki|}}}Petrie polygon|Petrie polygon]]s{{Efn|{{#if:{{{instance|}}}||Coxeter describes the helical Petrie polygons of regular 4-polytopes. He begins by noting that the regular tesselations of 3-space (which may be viewed as &quot;flat&quot; 4-polytopes) have the same kind of helical Petrie polygons as spherical 4-polytopes:&lt;blockquote&gt;Among the vertices and edges of a regular honeycomb &lt;math&gt;\{p, q, r\}&lt;/math&gt; we can pick out a new kind of ''Petrie polygon'' in which every three consecutive edges belong to the Petrie polygon of a cell but no four consecutive edges belong to the same cell. ... The isometry which takes us one step along the Petrie polygon, being conjugate to the product of half-turns about two opposite edges of the characteristic tetrahedron, is the product of half-turns about two skew lines, that is, a ''twist'': the product of a translation along a line &lt;math&gt;l&lt;/math&gt; (which measures the shortest distance between two skew lines) and a rotation about the same line. Thus the Petrie polygon is a &quot;helical&quot; polygon: its edges are the chords of a helix. This description is valid in hyperbolic space as well as in Euclidean space.
&lt;br&gt;In spherical space, &lt;math&gt;l&lt;/math&gt; is, of course, a great circle, the &quot;translation&quot; along it is a rotation about a polar great circle, and the twist is a ''compound rotation'' [double rotation]: the product of two rotations whose axes are polar great circles (lying in completely orthogonal planes of the Euclidean 4-space). Let &lt;math&gt;h&lt;/math&gt; denote the period of this compound rotation, so that the Petrie polygon is a skew &lt;math&gt;h&lt;/math&gt;-gon.{{Sfn|Coxeter|1970|p=25|loc=''Twisted Honeycombs'', §11. The Petrie polygon of a honeycomb}}&lt;/blockquote&gt;}}|name=Petrie polygon of a honeycomb}}
|1 [[5-cell#Boerdijk–Coxeter helix|pentagon]] x 2
|1 [[16-cell#Helical construction|octagon]] x 3
|2 [[{{{wiki|}}}Octagon#Skew octagon|octagon]]s x 4
|2 [[{{{wiki|}}}Dodecagon#Skew dodecagon|dodecagon]]s x 4
|4 [[{{{wiki|}}}30-gon#Petrie polygons|30-gon]]s x 6{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}11 [[{{{wiki|}}}Hendecagon#Related figures|{11/3}-gram]]s x 11
|}}}}
|20 [[{{{wiki|}}}30-gon#Petrie polygons|30-gon]]s x 4
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Edge length{{Efn|{{#if:{{{instance|}}}||A procedure to construct each of these 4-polytopes from the 4-polytope to its left (its predecessor) preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The successor edge length will always be less unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the ''same'' as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.}}|name=edge length of successor|group=}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{5}{2}} \approx 1.581&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{2-\phi} \approx 0.618&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{\phi^2}} \approx 0.874&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;}}}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2\phi^4}} \approx 0.270&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{\phi^4}} \approx 0.382&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Isocline chord{{Efn|{{#if:{{{instance|}}}||The isocline chord length is the 4-space distance by which each vertex is displaced in each step of the characteristic isoclinic (equi-angled) double rotation. Each vertex is displaced within its moving invariant central plane by a 2-space distance of one edge length, and displaced in 4-space by one isocline chord length along a circular, helical geodesic isocline. The invariant rotation planes are a Clifford parallel subset of all the central planes containing edges, but the subset captures all the vertices. Every edge is displaced to a parallel edge that lies the characteristic isocline chord distance away, whether or not the edge lies in an invariant plane during the rotation.}}|name=isocline chord|group=}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{2}} \approx 1.225&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{3} \approx 1.732&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{3} \approx 1.732&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{6} \approx 2.449&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{3} \approx 1.732&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{6} \approx 2.449&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2} \approx 1.414&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;}}}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{\phi^2}} \approx 0.618&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Isoclinic ratio{{Efn|{{#if:{{{instance|}}}||The ratio of the isocline chord to the edge length is a characteristic constant independent of the metric unit (long radius).}}|name=isocline chord to edge length ratio|group=}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{5}} \approx 0.775&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \approx 1.732&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \approx 1.732&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi+1} \approx 1.618&lt;/math&gt;&lt;/small&gt;{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|}}}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi+1} \approx 1.618&lt;/math&gt;&lt;/small&gt;
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|{{#ifeq:{{{radius|}}}|1|[[24-cell#Great hexagons|Long radius]]|{{#ifeq:{{{radius|}}}|{{radic|2}}|[[24-cell#Great squares|Long radius]]|Long radius}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Edge radius
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{8}} \approx 0.612&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{3} \approx 1.732&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{2}} \approx 1.225&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi\sqrt{5}}{4}} \approx 0.951&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi\sqrt{5}}{4}} \approx 1.345&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;}}}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3\phi^2}{8}} \approx 0.991&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3\phi^2}{4}} \approx 1.401&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Face radius
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{25}{86}} \approx 0.539&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{25}{43}} \approx 0.762&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{4}{3}} \approx 1.155&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{5\phi^2}{46}} \approx 0.533&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{5\phi^2}{23}} \approx 0.754&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;}}}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^3}{8\sqrt{5/2}}} \approx 0.579&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^3}{4\sqrt{5/2}}} \approx 0.818&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Short radius{{Efn|Cell radius measured to the center of each cell, the vertices of the 4-polytope's dual polytope.}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{16}} = 0.25&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{8}} \approx 0.354&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}} \approx 0.926&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{4}} \approx 1.309&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;}}}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}} \approx 0.926&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{4}} \approx 1.309&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Area
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;10\left(\tfrac{5\sqrt{3}}{8}\right) \approx 10.825&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;10\left(\tfrac{5\sqrt{3}}{4}\right) \approx 21.651&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;32\left(\sqrt{\tfrac{3}{4}}\right) \approx 27.713&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;32\left(\sqrt{3}\right) \approx 55.425&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;24&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;48&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;96\left(\sqrt{\tfrac{3}{16}}\right) \approx 41.569&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;96\left(\sqrt{\tfrac{3}{4}}\right) \approx 83.138&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1200\left(\tfrac{\sqrt{3}}{4\phi^2}\right) \approx 198.48&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;1200\left(\tfrac{2\sqrt{3}}{4\phi^2}\right) \approx 396.95&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;720\left(\tfrac{\sqrt{25+10\sqrt{5}}}{8\phi^4}\right) \approx 90.366&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;720\left(\tfrac{\sqrt{25+10\sqrt{5}}}{4\phi^4}\right) \approx 180.73&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|Volume
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;5\left(\tfrac{5\sqrt{5}}{24}\right) \approx 2.329&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;5\left(\tfrac{5\sqrt{10}}{12}\right) \approx 6.588&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;16\left(\tfrac{1}{3}\right) \approx 5.333&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;16\left(\tfrac{2\sqrt{2}}{3}\right) \approx 15.085&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;8&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;8\sqrt{8} \approx 22.627&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;24\left(\tfrac{\sqrt{2}}{3}\right) \approx 11.314&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;24\left(\tfrac{4}{3}\right) = 32&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;600\left(\tfrac{\sqrt{2}}{12\phi^3}\right) \approx 16.693&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;600\left(\tfrac{4}{12\phi^3}\right) \approx 47.214&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;120\left(\tfrac{15 + 7\sqrt{5}}{4\phi^6\sqrt{8}}\right) \approx 18.118&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;120\left(\tfrac{15 + 7\sqrt{5}}{4\phi^6}\right) \approx 51.246&lt;/math&gt;&lt;/small&gt;}}}}
|-
!style=&quot;vertical-align:top;text-align:right;&quot;|4-Content
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5}}{24}\left(\tfrac{\sqrt{5}}{2}\right)^4 \approx 0.146&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5}}{24}\left(\sqrt{5}\right)^4 \approx 2.329&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{2}{3} \approx 0.667&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{8}{3} \approx 2.666&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;8&lt;/math&gt;&lt;/small&gt;}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 3.863&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 15.451&lt;/math&gt;&lt;/small&gt;}}}}{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
{{#ifeq:{{{columns|7}}}|7||{{#ifeq:{{{columns}}}|9|
{{!}}
|}}}}
|{{#ifeq:{{{radius|1}}}|1|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 4.193&lt;/math&gt;&lt;/small&gt;|{{#ifeq:{{{radius}}}|{{radic|2}}|&lt;small&gt;&lt;math&gt;\tfrac{\text{Short}\times\text{Vol}}{4} \approx 16.770&lt;/math&gt;&lt;/small&gt;}}}}
|}&lt;noinclude&gt;
{{#if:{{{wiki|}}}|{{Polyscheme}}|}}
{{Notelist}}
{{Reflist}}
[[{{{wiki|}}}Category:Geometry templates]]
{{#if:{{{wiki|}}}|[[Category:Polyscheme]]|}}
&lt;/noinclude&gt;</text>
      <sha1>jmqd0aupwz1l9ai3xmace3ysozvsoe7</sha1>
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  </page>
  <page>
    <title>User:Platos Cave (physics)/Simulation Hypothesis/Planck units (geometrical)</title>
    <ns>2</ns>
    <id>275012</id>
    <revision>
      <id>2810705</id>
      <parentid>2785980</parentid>
      <timestamp>2026-05-21T01:40:07Z</timestamp>
      <contributor>
        <username>Platos Cave (physics)</username>
        <id>2562653</id>
      </contributor>
      <origin>2810705</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35345" sha1="a63e902johdgbbvjbiaxg9156o9ckvr" xml:space="preserve">{{Original research}}

'''Natural Planck units as geometrical objects (the mathematical electron model)'''

In the [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical) |mathematical electron]] &lt;ref&gt;Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}&lt;/ref&gt; model, the electron is assigned a geometrical formula ψ, the formula itself the geometry of 2 dimensionless constants (α, Ω) and π, and resembles the formula for the volume of a torus or surface of a 4-axis hypersphere. Embedded within this formula ψ are geometrical analogues MTP of the [[w:Planck units |Planck units]] for [[w:Planck mass |Planck mass]], [[w:Planck time |Planck time]] and [[w:Planck momentum |Planck momentum]] where M = 1, T = π, P = Ω. From these 3 Planck objects and a base-15 guide-rail can be reconstructed the dimensionless physical constants ''G'', ''h'', ''c'', ''e'', (''y''&lt;sub&gt;e&lt;/sub&gt;/''g''&lt;sub&gt;e&lt;/sub&gt;), ''m''&lt;sub&gt;e&lt;/sub&gt;, ''k''&lt;sub&gt;B&lt;/sub&gt;.

Geometrical objects are chosen over numerical unit systems as the object embeds the attribute (mass, length, time, charge) within its geometry (i.e.: the object for length embeds the function of length), whereas for numerical based constants, the numerical values are dimensionless frequencies of the unit system, for example (in [[w:SI units |SI units]]) 3m refers to 3 of the unit m, the number 3 carries no length-specific information.

For a statistical analysis of this page &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=6. Physical Constant Anomalies as Evidence of a Mathematical Universe |journal=RG |date=Dec 2021 | doi=10.13140/RG.2.2.15874.15041/9}}&lt;/ref&gt;&lt;ref&gt;https://codingthecosmos.com/ Statistical analysis of the mathematical electron&lt;/ref&gt;


=== Geometrical objects ===
The principle constants are the (physical) [[w:fine-structure constant | fine structure constant '''α''']] and the mathematical constants '''Ω''' and '''π'''. Omega itself is the geometry of π and Euler's number [[w:E_(mathematical_constant) |e]] = 2.718281828459...;

:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;

The fine structure constant alpha can be derived via this model and so here is assigned the letter ''a'' = 137.03599... to represent the analogue of the inverse fine structure α&lt;sup&gt;-1&lt;/sup&gt; = 137.03599...

From MTPα we can derive further Planck unit analogues length L and charge A. As geometrical objects, we also have the option to interlock them [[w:Lego |Lego]] style. For this purpose we may identify a unit number relationship θ that dictates how the objects may '''fit''' together (i.e.: we can combine the objects for length L and time T to form the object for velocity V = L/T). 

{| class=&quot;wikitable&quot;
|+Table 2. MLTVA Geometrical objects
! attribute
! geometrical object
! unit number θ
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;
| 15
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;
| -30
|-
| [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| 16
|-
| velocity  
| &lt;math&gt;V = \frac{2\pi P^2}{M} = (2\pi\Omega^2)&lt;/math&gt;
| 17
|-
| length  
| &lt;math&gt;L = VT = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13
|-
| ampere 
| &lt;math&gt;A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| 3
|-
| temperature 
| &lt;math&gt;K = \frac{AV}{2\pi} = (\frac{2^7 \pi^3 \Omega^5}{a})&lt;/math&gt;
| 20
|}


As the geometries of dimensionless constants, these objects are also dimensionless and so are independent of any system of units, and of any numerical system, and so could qualify as &quot;natural units&quot; (naturally occuring units);
{{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...''
...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as &quot;natural units&quot;... -Max Planck &lt;ref&gt;Planck (1899), p. 479.&lt;/ref&gt;&lt;ref name=&quot;TOM&quot;&gt;*Tomilin, K. A., 1999, &quot;[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]&quot;, 287–296.&lt;/ref&gt;}}  


==== Scalars ==== 

To translate from geometrical objects to a numerical system of units requires system dependent scalars ('''kltpva'''). For example; 

:If we use ''k'' to convert ''M'' (M=1) to the SI Planck mass (M*''k''&lt;sub&gt;SI&lt;/sub&gt; = &lt;math&gt;m_P&lt;/math&gt;), then ''k''&lt;sub&gt;SI&lt;/sub&gt; = 0.2176728e-7kg ([[w:SI_units |SI units]])

:''c'' = V*''v''&lt;sub&gt;SI&lt;/sub&gt; = 299792458m/s ([[w:SI_units |SI units]])

:''c'' = V*''v''&lt;sub&gt;imp&lt;/sub&gt; = 186282miles/s ([[w:Imperial_units |imperial units]])


==== Scalar relationships ====
Scalars that translate to the SI unit system must therefore carry not only the numerical conversion but also the unit (as MTP themselves are unitless), i.e.: scalar ''v'' = 11843707.905 m/s. This also means that the scalars follow the unit number relationship which we can also denote as ''u''&lt;sup&gt;θ&lt;/sup&gt;, and so we can find ratios where the scalars cancel. 

{| class=&quot;wikitable&quot;
|+Table 5. Geometrical units
! Attribute
! Geometrical object
! Scalar 
! Unit ''u''&lt;sup&gt;θ&lt;/sup&gt;
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;  
| ''k''
| &lt;math&gt;u^{15}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;  
| ''t''
| &lt;math&gt;u^{-30}&lt;/math&gt;
|-
| [[v:User:Platos Cave (physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| ''r''&lt;sup&gt;2&lt;/sup&gt;
| &lt;math&gt;u^{16}&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;  
| ''v''
| &lt;math&gt;u^{17}&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| ''l''
| &lt;math&gt;u^{-13}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| ''q''
| &lt;math&gt;u^3&lt;/math&gt;
|}


Here are examples (units = 1), as such ''only 2 scalars are required'', for example, if we know the numerical value for ''q'' and for ''l'' then we know the numerical value for ''t'' ('''t = q&lt;sup&gt;3&lt;/sup&gt;l&lt;sup&gt;3&lt;/sup&gt;'''), and from ''l'' and ''t'' we know the value for ''k''. 

:&lt;math&gt;\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{q^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1&lt;/math&gt;


In other words, once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, [[w:permeability of vacuum|μ&lt;sub&gt;0&lt;/sub&gt;]]) are assigned exact values, following the [[w:2019 redefinition of SI base units|2019 redefinition of SI base units]] a total of 4 constants now have independently exact values assigned which is problematic in terms of this model. 

Scalars ''r'' (θ = 8) and ''v'' (θ = 17) are chosen for demonstration as they can be derived directly from the 2 constants with exact values; ''c'' and ''μ&lt;sub&gt;0&lt;/sub&gt;''.


{| class=&quot;wikitable&quot;
|+Table 6. Geometrical objects
! attribute
! geometrical object
! unit number θ
! scalar r(8), v(17)
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;
| 15 = 8*4-17
| &lt;math&gt;k = \frac{r^4}{v}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;
|  -30 = 8*9-17*6 
| &lt;math&gt;t = \frac{r^9}{v^6}&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;
| 17
| ''v''
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13 = 8*9-17*5
| &lt;math&gt;l = \frac{r^9}{v^5}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})&lt;/math&gt; 
| 3 = 17*3-8*6
| &lt;math&gt;q = \frac{v^3}{r^6}&lt;/math&gt;
|}


{| class=&quot;wikitable&quot;
|+ Table 7. Comparison; SI and θ
! constant
! θ (SI unit)
! MLTVA
! scalar r(8), v(17)
|-
| ''c'' 
| &lt;math&gt;\frac{m}{s}&lt;/math&gt; (-13+30 = {{font color|red|white|17}})
| ''c*'' = &lt;math&gt;V*v&lt;/math&gt; 
| {{font color|red|white|17}}
|-
| ''h'' 
| &lt;math&gt;\frac{kg \;m^2}{s}&lt;/math&gt; (15-26+30={{font color|red|white|19}})
| ''h*'' = &lt;math&gt;2 \pi M V L * \frac{r^{13}}{v^5}&lt;/math&gt; 
| 8*13-17*5={{font color|red|white|19}}
|-
| ''G''
| &lt;math&gt;\frac{m^3}{kg \;s^2}&lt;/math&gt; (-39-15+60={{font color|red|white|6}})
| ''G*'' = &lt;math&gt;\frac{V^2 L}{M} * \frac{r^5}{v^2}&lt;/math&gt; 
| 8*5-17*2={{font color|red|white|6}}
|-
| ''e'' 
| &lt;math&gt;C = A s&lt;/math&gt; (3-30={{font color|red|white|-27}})
| ''e*'' = &lt;math&gt;A T * \frac{r^3}{v^3}&lt;/math&gt; 
| 8*3-17*3={{font color|red|white|-27}}
|-
| ''k&lt;sub&gt;B&lt;/sub&gt;'' 
| &lt;math&gt;\frac{kg \;m^2}{s^2 \;K}&lt;/math&gt; (15-26+60-20={{font color|red|white|29}})
| ''k&lt;sub&gt;B&lt;/sub&gt;*'' = &lt;math&gt;\frac{2 \pi V M}{A} * \frac{r^{10}}{v^3}&lt;/math&gt; 
| 8*10-17*3={{font color|red|white|29}}
|-
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' 
| &lt;math&gt;\frac{kg \;m}{s^2 \;A^2}&lt;/math&gt; (15-13+60-6={{font color|red|white|56}})
| ''μ&lt;sub&gt;0&lt;/sub&gt;*'' = &lt;math&gt;\frac{4 \pi V^2 M}{a L A^2} * r^7&lt;/math&gt; 
| 8*7={{font color|red|white|56}}
|}


====CODATA 2014====
Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k&lt;sub&gt;B&lt;/sub&gt;''), consequently here we are using CODATA 2014 values. This is because only 2 dimensioned physical constants can be assigned exact values, once 2 constants have been assigned values, then all other constants are defined by default. In CODATA 2014 2 constants have exact values; &lt;math&gt;c&lt;/math&gt; and the [[w:Vacuum permeability | vacuum permeability]] &lt;math&gt;\mu_0&lt;/math&gt;. 

:&lt;math&gt;c = 299792458&lt;/math&gt; m/s
:&lt;math&gt;\mu_0 = 4\pi / 10^7&lt;/math&gt;

:&lt;math&gt;v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;

:&lt;math&gt;r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{a};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;


==== Fine structure constant ====
Classically the fine structure constant can be expressed by this formula.

:&lt;math&gt;\frac{2 h}{\mu_0 e^2 c} = \color{red}\alpha^{-1} \color{black}&lt;/math&gt;


If we insert the geometrical analogues alpha emerges, units and scalars cancel, validating the unit number relationship and geometries.

:&lt;math&gt;\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{a}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{a})^2(2 \pi \Omega^2) = 
 \color{red}a \color{black}&lt;/math&gt;

:&lt;math&gt;units \;\frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1&lt;/math&gt;

:&lt;math&gt;scalars \;(\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1&lt;/math&gt;

Thus proving that &lt;math&gt; \color{red}\alpha\color{black} = \color{red}\alpha \color{black}&lt;/math&gt;


==== Electron formula ====
{{main|User:Platos Cave (physics)/Simulation_Hypothesis/Electron (mathematical)}}
The ''electron object'' (''formula ψ'') is a mathematical particle (units and scalars cancel). 

:&lt;math&gt;\psi = 4\pi^2(2^6 3 \pi^2 a \Omega^5)^3 = .23895453...x10^{23}&lt;/math&gt; units = 1


In this example, embedded within the electron are the objects for charge, length and time ALT. AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]].

:&lt;math&gt;T = \pi \frac{r^9}{v^6},\; u^{-30}&lt;/math&gt;

:&lt;math&gt;\sigma_{e} = \frac{3 a^2 A L}{2\pi^2} = {2^7 3 \pi^3 a \Omega^5}\frac{r^3}{v^2},\; u^{-10}&lt;/math&gt;

:&lt;math&gt;\psi = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 a \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1&lt;/math&gt;


Associated with the electron are dimensioned parameters, these parameters however are a function of the MLTA units, the formula ψ dictating the frequency of these units. By setting MLTA to their SI Planck unit equivalents (Table 6.);

[[w:electron mass | electron mass]]     &lt;math&gt;m_e^* = \frac{M}{\psi}&lt;/math&gt; (M = [[w:Planck mass | Planck mass]] = &lt;math&gt;\frac{r^4}{v})&lt;/math&gt; 

[[w:Compton wavelength | electron wavelength]] &lt;math&gt;\lambda_e^* = 2\pi L \psi&lt;/math&gt; (L = [[w:Planck length | Planck length]] = &lt;math&gt;2\pi\Omega^2\frac{r^9}{v^5})&lt;/math&gt;

[[w:elementary charge | elementary charge]]    &lt;math&gt;e^* = A\;T&lt;/math&gt; (T = [[w:Planck time | Planck time]]) = &lt;math&gt;\frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3}&lt;/math&gt; 

[[w:Rydberg constant | Rydberg constant]] &lt;math&gt;R^* = (\frac{m_e}{4 \pi L a^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} a^5 \Omega^{17}}\frac{v^5}{r^9}\;u^{13}&lt;/math&gt; 


==== Omega ====
The most precise of the experimentally measured constants is the [[w:Rydberg constant | Rydberg constant]] ''R'' = 10973731.568508(65) 1/m. Here ''c'' (exact),  [[w:Vacuum permeability | Vacuum permeability]] μ&lt;sub&gt;0&lt;/sub&gt; = 4π/10^7 (exact) and  ''R'' (12-13 digits) are combined into a unit-less ratio;

:&lt;math&gt;\mu_0^* = \frac{4 \pi V^2 M}{a L A^2} = \frac{a}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}&lt;/math&gt;

:&lt;math&gt;R^* = (\frac{m_e}{4 \pi L a^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} a^5 \Omega^{17}}  \frac{v^5}{r^9},\;u^{13}&lt;/math&gt;


:&lt;math&gt;\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{a}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1}{2^{23} 3^3 \pi^{11} a^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7}&lt;/math&gt;

:&lt;math&gt;\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = 2^{295} \pi^{157} 3^{21} a^{26} (\Omega^{15})^{15}&lt;/math&gt;, units = 1


We can now define ''Ω'' using the geometries for (''c&lt;sup&gt;*&lt;/sup&gt;, μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;, R&lt;sup&gt;*&lt;/sup&gt;'') and then solve by replacing (''c&lt;sup&gt;*&lt;/sup&gt;, μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;, R&lt;sup&gt;*&lt;/sup&gt;'') with the numerical (''c, μ&lt;sub&gt;0&lt;/sub&gt;, R''). 

:&lt;math&gt;\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 a^{26}}, \;units = 1&lt;/math&gt;

:&lt;math&gt;\Omega = 2.007\;134\;949\;636...,\; units = 1&lt;/math&gt; (CODATA 2014 mean values)

:&lt;math&gt;\Omega = 2.007\;134\;949\;687...,\; units = 1&lt;/math&gt; (CODATA 2018 mean values)


There is a natural number for Ω that is a square root implying that Ω can have a plus or a minus solution, and this agrees with theory (in the mass domain Ω occurs as Ω&lt;sup&gt;2&lt;/sup&gt; = plus only, in the charge domain Ω occurs as Ω&lt;sup&gt;3&lt;/sup&gt; = can be plus or minus; see [[v:Sqrt_Planck_momentum | sqrt(momentum)]]). This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants).

:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;

Using this Omega and reversing the above formula solves α = 137.035996376

We may also consider including [[w:Euler%27s_formula | Euler's_formula]] where {{mvar|i}} is the [[w:imaginary unit | imaginary unit]].

:&lt;math display=&quot;block&quot;&gt;e^{i x} = \cos x + i \sin x&lt;/math&gt;


==== Dimensionless combinations ====
According to the unit number relationship, we can combine the physical constants in combinations where the unit numbers cancel, in this model these combinations are dimensionless, however they still retain SI units. If the model is correct (if the combinations are dimensionless) then the scalars will also have cancelled and numerically the solutions using CODATA or Geometrical objects will approach equality (barring uncertainties).These combinations can be used to test the veracity of the MLTA geometries as natural Planck units.


Example:

:&lt;math&gt;\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = &lt;/math&gt; {{font color|green|yellow| '''0.228 473 759... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}

:&lt;math&gt;\frac{h^3}{e^{13} c^{24}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 639... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}, units = &lt;math&gt;\frac{kg^3 s^8}{m^{18} A^{13}}&lt;/math&gt;, units = 1 (15*3-30*8+13*18-3*13 = 0)

 Note: the geometry &lt;math&gt;\color{red}(\Omega^{15})^n\color{black}&lt;/math&gt; (integer n ≥ 0) is common to all ratios where units and scalars cancel (i.e.: only combinations with &lt;math&gt;\Omega^0, \Omega^{15}, \Omega^{30}, \Omega^{45}&lt;/math&gt;... will be dimensionless). However there is no Planck unit with a &lt;math&gt;\Omega^{15}&lt;/math&gt; component (all constants are combinations of &lt;math&gt;\Omega^2&lt;/math&gt; and &lt;math&gt;\Omega^3&lt;/math&gt;), and this suggests there is an underlying geometrical base-15.


{| class=&quot;wikitable&quot;
|+Table 8. Dimensionless combinations
! CODATA 2014 mean
! (α, Ω) mean
! units = 1
! scalars = 1
|-
| &lt;math&gt;\frac{k_B e c}{h} =&lt;/math&gt; {{font color|green|yellow|'''1.000 8254'''}}
| &lt;math&gt;\frac{(k_B^*) (e^*) (c^*)}{(h^*)}&lt;/math&gt; = {{font color|green|yellow|'''1.0'''}}
| &lt;math&gt;\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1&lt;/math&gt;
|-
| &lt;math&gt;\frac{h^3}{e^{13} c^{24}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 639... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 759... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1&lt;/math&gt;
|-
| &lt;math&gt;\frac{c^9 e^4}{m_e^3} =&lt;/math&gt; {{font color|green|yellow| '''0.170 514 342... 10&lt;sup&gt;92&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black}=&lt;/math&gt; {{font color|green|yellow| '''0.170 514 368... 10&lt;sup&gt;92&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1&lt;/math&gt;
| &lt;math&gt;(v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1&lt;/math&gt; 
|-
| &lt;math&gt;\frac{k_B}{e^2 m_e c^4} =&lt;/math&gt; {{font color|green|yellow| '''73 095 507 858.'''}}
| &lt;math&gt;\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} =&lt;/math&gt; {{font color|green|yellow| '''73 035 235 897.'''}} 
| &lt;math&gt;\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1&lt;/math&gt;
|}


==== Derivation via CODATA ====
As geometrical objects, the physical constants (''G, h, e, m&lt;sub&gt;e&lt;/sub&gt;, k&lt;sub&gt;B&lt;/sub&gt;'') can also be defined using the geometrical formulas for (''c&lt;sup&gt;*&lt;/sup&gt;, μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;, R&lt;sup&gt;*&lt;/sup&gt;'') and solved using the numerical (mean) values for (''c, μ&lt;sub&gt;0&lt;/sub&gt;, R, α''). For example;

:&lt;math&gt;{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{3^{19} \pi^{12} \Omega^{12} r^{39} u^{57}}{v^{15}},\; \theta = 57&lt;/math&gt; ... '''and''' ... 

:&lt;math&gt;\frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2} = \frac{3^{19} \pi^{12} \Omega^{12} r^{39} u^{57}}{v^{15}},\; \theta = 57&lt;/math&gt;


{| class=&quot;wikitable&quot;
|+Table 12. Calculated from (R, c, μ&lt;sub&gt;0&lt;/sub&gt;, α) columns 2, 3, 4 vs CODATA 2014 columns 5, 6 
! Constant
! Formula
! Units
! Calculated from (R, c, μ&lt;sub&gt;0&lt;/sub&gt;, α)
! CODATA 2014 &lt;ref&gt;[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)&lt;/ref&gt;
! Units
|-
| [[w:Planck constant | Planck constant]]
| &lt;math&gt;{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R}^2}&lt;/math&gt; 
| &lt;math&gt;\frac{kg^3}{A^6 s}&lt;/math&gt;, θ = 57
| ''h&lt;sup&gt;*&lt;/sup&gt;'' = 6.626 069 134 e-34, θ = 19 
| ''h'' = 6.626 070 040(81) e-34 
| &lt;math&gt;\frac{kg \;m^2}{s}&lt;/math&gt;, θ = 19
|-
| [[w:Gravitational constant | Gravitational constant]]
| &lt;math&gt;{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R}^2}&lt;/math&gt; 
| &lt;math&gt;\frac{kg\; m^3}{A^2 s^2}&lt;/math&gt;, θ = 30
| ''G&lt;sup&gt;*&lt;/sup&gt;'' = 6.672 497 192 29 e11, θ = 6
| ''G'' = 6.674 08(31) e-11
| &lt;math&gt;\frac{m^3}{kg \;s^2}&lt;/math&gt;, θ = 6
|-
| [[w:Elementary charge | Elementary charge]]
| &lt;math&gt;{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R}}&lt;/math&gt;
| &lt;math&gt;\frac{s^4}{A^3}&lt;/math&gt;, θ = -81
| ''e&lt;sup&gt;*&lt;/sup&gt;'' = 1.602 176 511 30 e-19, θ = -27
| ''e'' = 1.602 176 620 8(98) e-19
| &lt;math&gt;A s&lt;/math&gt;, θ = -27
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| &lt;math&gt;{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R}}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3}{s^2 A^6}&lt;/math&gt;, θ = 87
| ''k&lt;sub&gt;B&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 1.379 510 147 52 e-23, θ = 29
| ''k&lt;sub&gt;B&lt;/sub&gt;'' = 1.380 648 52(79) e-23
| &lt;math&gt;\frac{kg \;m^2}{s^2 \;K}&lt;/math&gt;, θ = 29
|-
| [[w:Electron mass | Electron mass]]
| &lt;math&gt;{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 \alpha^7}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3 s^2}{m^6 A^6}&lt;/math&gt;, θ = 45
| '' m&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 9.109 382 312 56 e-31, θ = 15
| ''m&lt;sub&gt;e&lt;/sub&gt;'' = 9.109 383 56(11) e-31 
| &lt;math&gt;kg&lt;/math&gt;, θ = 15
|-
| [[w:Gyromagnetic ratio | Gyromagnetic ratio]]
| &lt;math&gt;({(\gamma_e^*)/2\pi})^3 = \frac{g_e^3 3^3 c^4}{2^8 \pi^8 \alpha \mu_0^3 R_\infty^2}&lt;/math&gt;
| &lt;math&gt;\frac{m^3 s^2 A^6}{kg^3}&lt;/math&gt;, θ = -126
| ''(γ&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;/2π)'' = 28024.953 55, θ = -42
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π'' = 28024.951 64(17) 
| &lt;math&gt;\frac{A\;s}{kg}&lt;/math&gt;, θ = -42
|-
| [[w:Planck mass | Planck mass]]
| &lt;math&gt;({m_P^*})^{15} = \frac{2^{25} \pi^{13} {\mu_0}^6}{3^6 c^5 \alpha^{16} R^2}&lt;/math&gt;
| &lt;math&gt;\frac{kg^6 m^3}{s^7 A^{12}}&lt;/math&gt;, θ = 225
| ''m&lt;sub&gt;P&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 0.217 672 817 580 e-7, θ = 15
| ''m&lt;sub&gt;P&lt;/sub&gt;'' = 0.217 647 0(51) e-7
| &lt;math&gt;kg&lt;/math&gt;, θ = 15
|-
| [[w:Planck length | Planck length]]
| &lt;math&gt;({l_p^*})^{15} = \frac{\pi^{22} {\mu_0}^9}{2^{35} 3^{24} \alpha^{49} c^{35} R^8}&lt;/math&gt;
| &lt;math&gt;\frac{kg^9 s^{17}}{m^{18}A^{18}}&lt;/math&gt;, θ = -195
| ''l&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 0.161 603 660 096 e-34, θ = -13
| ''l&lt;sub&gt;p&lt;/sub&gt;'' =  0.161 622 9(38) e-34
| &lt;math&gt;m&lt;/math&gt;, θ = -13
|}


==== Base-15 geometry ====
We can construct a table of constants using these 3 geometries. Setting a dimensionless (units = scalars = 0) conversion factor f(x);

:&lt;math&gt;f(x)\;units = (\frac{L^{15}}{M^9 T^{11}})^n = 1&lt;/math&gt; θ = (-13*15) - (15*9) - (-30*11) = 0


:&lt;math&gt;\color{red}i\color{black} = \pi^2 \Omega^{15}&lt;/math&gt;, units = &lt;math&gt;\sqrt{f(x)}&lt;/math&gt; = 1 (unit number = 0, no scalars)

:&lt;math&gt;\color{red}x\color{black} = \Omega \frac{v}{r^2}&lt;/math&gt; , units = &lt;math&gt;\sqrt{\frac{L}{M T}}&lt;/math&gt; = u&lt;sup&gt;1&lt;/sup&gt; = u (unit number = -13 -15 +30 = 2/2 = 1, with scalars ''v'', ''r'')

:&lt;math&gt;\color{red}y\color{black} = \pi \frac{r^{17}}{v^8}&lt;/math&gt; , units = &lt;math&gt;M^2 T&lt;/math&gt; = 1, (unit number = 15*2 -30 = 0, with scalars ''v'', ''r'') 


Note: The following suggests a numerical boundary to the values the SI constants can have.

:&lt;math&gt;\frac{v}{r^2} = a^{1/3} = \frac{1}{t^{2/15}k^{1/5}} = \frac{\sqrt{v}}{\sqrt{k}}&lt;/math&gt; ... = 23326079.1...; unit = u

:&lt;math&gt;\frac{r^{17}}{v^8} = k^2 t = \frac{k^{17/4}}{v^{15/4}} = ... &lt;/math&gt; gives a range from 0.812997... x10&lt;sup&gt;-59&lt;/sup&gt; to 0.123... x10&lt;sup&gt;60&lt;/sup&gt;


Note: Influence of &lt;math&gt;f(x)&lt;/math&gt;, units = 1

:&lt;math&gt;\frac{r^{17}}{v^8} \;\;units \;(\frac{M^2 L^8}{T^7}) (\frac{T}{L})^8  = M^2 T&lt;/math&gt;

:&lt;math&gt;r^{17} \;\;units \;(\frac{M\;L}{T})^{17/4} fx^{1/4} = \frac{M^2\;L^8}{T^7}&lt;/math&gt;

:&lt;math&gt;r \;\;units \;(\frac{M\;L}{T})^{1/4} fx^{1/4} = \frac{L^4}{M^2 T^3}&lt;/math&gt;


{| class=&quot;wikitable&quot;
|+Table 9. Table of Constants
! Constant
! θ
! Geometrical object (α, Ω, v, r)
! Unit
! Calculated
! CODATA 2014
|-
| Time (Planck)
| &lt;math&gt;\color{red}-30\color{black}&lt;/math&gt;
| &lt;math&gt;T = \color{red}\frac{x^\theta i^2}{y^3}\color{black} = \frac{\pi r^9}{v^6}&lt;/math&gt;  
| &lt;math&gt;T&lt;/math&gt;
| T = 5.390 517 866 e-44
| ''t&lt;sub&gt;p&lt;/sub&gt;'' = 5.391 247(60) e-44
|-
| [[w:Elementary charge |Elementary charge]]  
| &lt;math&gt;\color{red}-27\color{black}&lt;/math&gt;
| &lt;math&gt;e^* = (\frac{2^7 \pi^3}{\alpha}) \color{red}\frac{x^\theta i^2}{y^3}\color{black} = (\frac{2^7 \pi^3}{\alpha}) \;\frac{\pi \Omega^3 r^3}{v^3}&lt;/math&gt; 
| &lt;math&gt;\frac{L^{3/2}}{T^{1/2} M^{3/2}} = AT&lt;/math&gt;
| ''e&lt;sup&gt;*&lt;/sup&gt;'' = 1.602 176 511 30 e-19
| ''e'' = 1.602 176 620 8(98) e-19
|-
| Length (Planck) 
| &lt;math&gt;\color{red}-13\color{black}&lt;/math&gt;
| &lt;math&gt;L = (2\pi) \color{red}\frac{x^\theta i}{y}\color{black} = (2\pi) \;\frac{\pi \Omega^2 r^9}{v^5}&lt;/math&gt; 
| &lt;math&gt;L&lt;/math&gt;
| L = 0.161 603 660 096 e-34
| ''l&lt;sub&gt;p&lt;/sub&gt;'' = 0.161 622 9(38) e-34
|-
| Ampere 
| &lt;math&gt;\color{red}3\color{black}&lt;/math&gt;
| &lt;math&gt;A = (\frac{2^7 \pi^3}{\alpha}) \color{red}x^\theta\color{black} = (\frac{2^7 \pi^3}{\alpha}) \; \frac{\Omega^3 v^3}{r^6} &lt;/math&gt; 
| &lt;math&gt;A = \frac{L^{3/2}}{M^{3/2} T^{3/2}}&lt;/math&gt;
| A = 0.297 221 e25
| ''e/t&lt;sub&gt;p&lt;/sub&gt;'' = 0.297 181 e25
|-
| [[w:Gravitational constant |Gravitational constant]]  
| &lt;math&gt;\color{red}6\color{black}&lt;/math&gt;
| &lt;math&gt;G^* = (2^3 \pi^3) \color{red}\color{red}x^\theta y\color{black} = (2^3 \pi^3) \;\frac{\pi \Omega^6 r^5}{v^2}&lt;/math&gt; 
| &lt;math&gt;\frac{L^3}{M T^2}&lt;/math&gt;
| ''G&lt;sup&gt;*&lt;/sup&gt;'' =  6.672 497 192 29 e11
| ''G'' = 6.674 08(31) e-11
|-
|  
| &lt;math&gt;\color{red}8\color{black}&lt;/math&gt;
| &lt;math&gt;X = (2^4 \pi^4) \color{red}\color{red}x^\theta y\color{black} = (2^4 \pi^4) \pi \Omega^8 r&lt;/math&gt; 
| &lt;math&gt;\frac{L^4}{M^2 T^3}&lt;/math&gt;
| ''X'' =  918 977.554 22
| 
|-
| Mass (Planck)
| &lt;math&gt;\color{red}\color{red}15\color{black}&lt;/math&gt;
| &lt;math&gt;M = \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = \frac{r^4}{v}&lt;/math&gt;  
| &lt;math&gt;M&lt;/math&gt;
| M =  .217 672 817 580 e-7
| ''m&lt;sub&gt;P&lt;/sub&gt;'' = .217 647 0(51) e-7
|-
| [[v:User:Platos Cave (physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;\color{red}16\color{black}&lt;/math&gt;
| &lt;math&gt;P = \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = \Omega r^2&lt;/math&gt;
| &lt;math&gt;\frac{M^{1/2} L^{1/2}}{T^{1/2}}&lt;/math&gt;
| 
|
|-
| Velocity 
| &lt;math&gt;\color{red}\color{red}17\color{black}&lt;/math&gt;
| &lt;math&gt;V = (2\pi) \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = (2\pi) \;\Omega^2 v&lt;/math&gt;  
| &lt;math&gt;V = \frac{L}{T}&lt;/math&gt;
| V = 299 792 458
| ''c'' =  299 792 458
|-
| [[w:Planck constant |Planck constant]]  
| &lt;math&gt;\color{red}19\color{black}&lt;/math&gt;
| &lt;math&gt;h^* = (2^3 \pi^3) \color{red}\frac{x^\theta y^3}{i}\color{black} = (2^3 \pi^3) \;\frac{\pi \Omega^4 r^{13}}{v^5}&lt;/math&gt; 
| &lt;math&gt;\frac{L^2 M}{T}&lt;/math&gt;
| ''h&lt;sup&gt;*&lt;/sup&gt;'' = 6.626 069 134 e-34
| ''h'' = 6.626 070 040(81) e-34
|-
| [[w:Planck temperature |Planck temperature]]  
| &lt;math&gt;\color{red}\color{red}20\color{black}&lt;/math&gt;
| &lt;math&gt;{T_p}^* = (\frac{2^7 \pi^3}{\alpha}) \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = (\frac{2^7 \pi^3 }{\alpha}) \; \frac{\Omega^5 v^4}{r^6}&lt;/math&gt;  
| &lt;math&gt;\frac{L^{5/2}}{M^{3/2} T^{5/2}} = AV&lt;/math&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  1.418 145 219 e32
| ''T&lt;sub&gt;p&lt;/sub&gt;'' = 1.416 784(16) e32
|-
| [[w:Boltzmann constant |Boltzmann constant]]
| &lt;math&gt;\color{red}\color{red}29\color{black}&lt;/math&gt;
| &lt;math&gt;{k_B}^* = (\frac{\alpha}{2^5 \pi}) \color{red}\frac{x^\theta y^4}{i^2}\color{black} = (\frac{\alpha}{2^5 \pi }) \;\frac{r^{10}}{\Omega v^3}&lt;/math&gt;  
| &lt;math&gt;\frac{M^{5/2} T^{1/2}}{L^{1/2}} = \frac{M L}{T A}&lt;/math&gt;
| ''k&lt;sub&gt;B&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 1.379 510 147 52 e-23
| ''k&lt;sub&gt;B&lt;/sub&gt;'' = 1.380 648 52(79) e-23
|-
| [[w:Vacuum permeability |Vacuum permeability]] 
| &lt;math&gt;\color{red}56\color{black}&lt;/math&gt;
| &lt;math&gt;{\mu_0}^* = (\frac{\alpha}{2^{11} \pi^4}) \color{red}\frac{x^\theta y^7}{i^4}\color{black} = (\frac{\alpha}{2^{11} \pi^4})\; \frac{r^7}{\pi \Omega^4}&lt;/math&gt;  
| &lt;math&gt;\frac{M\;L}{T^2 A^2}&lt;/math&gt;
| ''μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 4π/10^7
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' = 4π/10^7
|}


==== Table of Constants ====
note: &lt;math&gt;\color{red}(u^{15})^n\color{black}&lt;/math&gt; constants have no Omega term.

{| class=&quot;wikitable&quot;
|+Table 10. Dimensioned constants; geometrical vs CODATA 2014
! Constant
! In Planck units
! Geometrical object
! SI calculated (r, v, Ω, α&lt;sup&gt;*&lt;/sup&gt;)
! SI CODATA 2014 &lt;ref&gt;[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)&lt;/ref&gt;
|-
| [[w:Speed of light | Speed of light]]
| V
|  &lt;math&gt;c^* = (2\pi\Omega^2)v,\;u^{17} &lt;/math&gt; 
| ''c&lt;sup&gt;*&lt;/sup&gt;'' = 299 792 458, unit = u&lt;sup&gt;17&lt;/sup&gt; 
| ''c'' =  299 792 458 (exact)
|-
| [[w:Fine structure constant | Fine structure constant]]
|  
| 
| ''α&lt;sup&gt;*&lt;/sup&gt;'' = 137.035 999 139 (mean)
| ''α'' = 137.035 999 139(31)
|-
| [[w:Rydberg constant | Rydberg constant]]
| &lt;math&gt;R^* = (\frac{m_e}{4 \pi L \alpha^2 M})&lt;/math&gt;  
| &lt;math&gt;R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} &lt;/math&gt; 
| ''R&lt;sup&gt;*&lt;/sup&gt;'' = 10 973 731.568 508, unit = u&lt;sup&gt;13&lt;/sup&gt;  
| ''R'' = 10 973 731.568 508(65)
|-
| [[w:Vacuum permeability | Vacuum permeability]]
| &lt;math&gt;\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}&lt;/math&gt; 
| &lt;math&gt;\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}&lt;/math&gt;
| ''μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 4π/10^7, unit = u&lt;sup&gt;56&lt;/sup&gt; 
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' = 4π/10^7 (exact)
|-
| [[w:Vacuum permittivity | Vacuum permittivity]]
| &lt;math&gt;\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}&lt;/math&gt; 
| &lt;math&gt;\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}&lt;/math&gt;
|  
|
|-
| [[w:Planck constant | Planck constant]]
| &lt;math&gt;h^* = 2 \pi M V L&lt;/math&gt;
| &lt;math&gt;h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}&lt;/math&gt;
| ''h&lt;sup&gt;*&lt;/sup&gt;'' = 6.626 069 134 e-34, unit = u&lt;sup&gt;19&lt;/sup&gt; 
| ''h'' = 6.626 070 040(81) e-34
|-
| [[w:Gravitational constant | Gravitational constant]]
| &lt;math&gt;G^* = \frac{V^2 L}{M}&lt;/math&gt;
| &lt;math&gt;G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{6}&lt;/math&gt; 
| ''G&lt;sup&gt;*&lt;/sup&gt;'' =  6.672 497 192 29 e11, unit = u&lt;sup&gt;6&lt;/sup&gt;
| ''G'' = 6.674 08(31) e-11
|-
| [[w:Elementary charge | Elementary charge]]
| &lt;math&gt;e^* = A T&lt;/math&gt;
| &lt;math&gt;e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{-27}&lt;/math&gt;
| ''e&lt;sup&gt;*&lt;/sup&gt;'' = 1.602 176 511 30 e-19, unit = u&lt;sup&gt;-27&lt;/sup&gt;
| ''e'' = 1.602 176 620 8(98) e-19
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| &lt;math&gt;k_B^* = \frac{2 \pi V M}{A}&lt;/math&gt; 
| &lt;math&gt;k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}&lt;/math&gt; 
| ''k&lt;sub&gt;B&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 1.379 510 147 52 e-23, unit = u&lt;sup&gt;29&lt;/sup&gt;
| ''k&lt;sub&gt;B&lt;/sub&gt;'' = 1.380 648 52(79) e-23
|-
| [[w:Electron mass | Electron mass]]
| 
| &lt;math&gt;m_e^* = \frac{M}{\psi},\; u^{15}&lt;/math&gt; 
| ''m&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 9.109 382 312 56 e-31, unit = u&lt;sup&gt;15&lt;/sup&gt;
| ''m&lt;sub&gt;e&lt;/sub&gt;'' = 9.109 383 56(11) e-31
|-
| [[w:Classical electron radius | Classical electron radius]]
|  
| &lt;math&gt;\lambda_e^* = 2\pi L \psi,\; u^{-13}&lt;/math&gt;  
| ''λ&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 2.426 310 2366 e-12, unit = u&lt;sup&gt;-13&lt;/sup&gt;
| ''λ&lt;sub&gt;e&lt;/sub&gt;'' = 2.426 310 236 7(11) e-12
|-
| [[w:Planck temperature | Planck temperature]]
| &lt;math&gt;T_p^* = \frac{A V}{\pi}&lt;/math&gt;
| &lt;math&gt;T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} &lt;/math&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  1.418 145 219 e32, unit = u&lt;sup&gt;20&lt;/sup&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;'' = 1.416 784(16) e32
|-
| [[w:Planck mass | Planck mass]]
| M
| &lt;math&gt;m_P^* = (1)\frac{r^4}{v} ,\; \color{red}\color{red}(u^{15})^1\color{black}&lt;/math&gt;
| ''m&lt;sub&gt;P&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  .217 672 817 580 e-7, unit = u&lt;sup&gt;15&lt;/sup&gt; 
| ''m&lt;sub&gt;P&lt;/sub&gt;'' = .217 647 0(51) e-7
|-
| [[w:Planck length | Planck length]]
| L
| &lt;math&gt;l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} &lt;/math&gt; 
| ''l&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = .161 603 660 096 e-34, unit = u&lt;sup&gt;-13&lt;/sup&gt;
| ''l&lt;sub&gt;p&lt;/sub&gt;'' = .161 622 9(38) e-34
|-
| [[w:Planck time | Planck time]]
| T
| &lt;math&gt;t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}\color{red}1/(u^{15})^2\color{black} &lt;/math&gt;
| ''t&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 5.390 517 866 e-44, unit = u&lt;sup&gt;-30&lt;/sup&gt;
| ''t&lt;sub&gt;p&lt;/sub&gt;'' = 5.391 247(60) e-44
|-
| [[w:Ampere | Ampere]]
| &lt;math&gt;A = \frac{16 V^3}{\alpha P^3}&lt;/math&gt;
| &lt;math&gt;A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 &lt;/math&gt;
| A&lt;sup&gt;*&lt;/sup&gt; = 0.297 221 e25, unit = u&lt;sup&gt;3&lt;/sup&gt;
| ''e/t&lt;sub&gt;p&lt;/sub&gt;'' = 0.297 181 e25 
|-
| [[w:Quantum Hall effect | Von Klitzing constant ]]
| &lt;math&gt;R_K^* = (\frac{h}{e^2})^*&lt;/math&gt;  
|  &lt;math&gt;R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}&lt;/math&gt; 
| ''R&lt;sub&gt;K&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 25812.807 455 59, unit = u&lt;sup&gt;73&lt;/sup&gt;
| ''R&lt;sub&gt;K&lt;/sub&gt;'' = 25812.807 455 5(59)
|-
| [[w:Gyromagnetic ratio | Gyromagnetic ratio]]
| 
| &lt;math&gt;\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}&lt;/math&gt;  
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π&lt;sup&gt;*&lt;/sup&gt;'' = 28024.953 55, unit = u&lt;sup&gt;-42&lt;/sup&gt;
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π'' = 28024.951 64(17) 
|}
 

==== Scalars (general)====

:&lt;math&gt;M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)&lt;/math&gt;

:&lt;math&gt;T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)&lt;/math&gt;

:&lt;math&gt;L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)&lt;/math&gt;

:&lt;math&gt;V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)&lt;/math&gt;

:&lt;math&gt;A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{\alpha})a = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)&lt;/math&gt;


===== MT to LPVA =====

In this example LPVA are derived from MT. The formulas for MT;

:&lt;math&gt;M = (1)k,\; unit = u^{15}&lt;/math&gt;

:&lt;math&gt;T = (\pi) t,\; unit = u^{-30}&lt;/math&gt;

Replacing scalars ''pvla'' with ''kt''

:&lt;math&gt;P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}&lt;/math&gt;

:&lt;math&gt;V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17}  &lt;/math&gt;

:&lt;math&gt;L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}&lt;/math&gt;

:&lt;math&gt;A = \frac{2^4 V^3}{\alpha P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{\alpha}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = 

u^{9/15*(-15)+6/15*30=3}    &lt;/math&gt;


===== PV to MTLA =====

In this example MLTA are derived from PV. The formulas for PV; 

:&lt;math&gt;P = (\Omega)p,\; unit = u^{16}&lt;/math&gt;

:&lt;math&gt;V = (2\pi\Omega^2)v,\; unit = u^{17}&lt;/math&gt;

Replacing scalars ''klta'' with ''pv''

:&lt;math&gt;M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} &lt;/math&gt;

:&lt;math&gt;T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} &lt;/math&gt;

:&lt;math&gt;L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}&lt;/math&gt;

:&lt;math&gt;A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}&lt;/math&gt;


===Geometry coded universe===
Modelling a Planck scale simulation universe using geometrical forms. For AI analysis, the model articles have been compiled into a single file &lt;ref&gt;https://theprogrammergod.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf Compilation of journal articles 1-8&lt;/ref&gt;.
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical)]]: Mathematical electron from Planck units 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass


=== External links === 
* [[w:Simulation_hypothesis | The Simulation hypothesis]]
* [https://codingthecosmos.com/ Statistical analysis of the mathematical electron]
* [https://theprogrammergod.com/ Overview of the mathematical electron model with links]
* [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article)


=== References === 
{{Reflist}}


[[Category: Physics]]
[[Category: Philosophy of science]]
__INDEX__</text>
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&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Bongo50 edits wikis!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

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      <text bytes="10349" sha1="1dp5eq6vtwb0di3s0v4tl2sml37ppv1" xml:space="preserve">This guide shows how to read the browsing history database created by the Mozilla Firefox web browser.
The title of this page can be understood as both &quot;browsing through the database&quot; and &quot;the database that contains the browsing history&quot;.

The database is named &lt;code&gt;places.sqlite&lt;/code&gt; and located in the [[Firefox#Profiles|profile folder]] and is in the SQL (''Structured Query Language'') 3 format. To facilitate using the commands, you may navigate to the profile folder directory, copy the database file into your working directory, or put the path to the database file in a variable.

== Simple searching ==
Some data can be viewed directly through text editors and command line tools, but they are inconveniently wrapped in a mess of non-human-readable binary data. 

When viewed through the command line simple checks for the existence of a title or URL can be performed using &lt;code&gt;grep -a -i -l &quot;string&quot; places.sqlite&lt;/code&gt; (parameters: search binary file, case insensitivity, list files only). Title searches can be performed using &lt;code&gt;grep -a -i -o -P &quot;.{30}string.{30}&quot; places.sqlite&lt;/code&gt;, where thirty characters around the string are also shown for context.

&lt;code&gt;grep -P &quot;[A-Za-z0-9/]&quot;&lt;/code&gt; can partially filter out junk characters for a &lt;code&gt;|less&lt;/code&gt; pipe. Additionally, &lt;code&gt;|sed -r &quot;s/http/\nhttp/g&quot;&lt;/code&gt; creates a line break before each URL.

== Listing the &quot;tables&quot; ==
First, we have to find the names of the &quot;tables&quot; that the database contains. These are the various areas within the database file that store parameters including the main history (&lt;code&gt;moz_places&lt;/code&gt;), bookmarks (&lt;code&gt;moz_bookmarks&lt;/code&gt;), favicons (&lt;code&gt;moz_favicons&lt;/code&gt;), and user-specified keywords (&lt;code&gt;moz_keywords&lt;/code&gt;).&lt;ref&gt;[https://renenyffenegger.ch/notes/development/web/browser/Firefox/profile-folder/places_sqlite/index ''renenyffenegger.ch'']&lt;/ref&gt;&lt;ref&gt;[https://stackoverflow.com/questions/947215/how-to-get-a-list-of-column-names-on-sqlite3-database ''How to get a list of column names on Sqlite3 database?'' – StackOverflow.com]&lt;/ref&gt;&lt;ref&gt;[https://www.sqlitetutorial.net/sqlite-tutorial/sqlite-show-tables/ ''SQLite Show Tables: Listing All Tables in a Database'' – SQLiteTutorial.net]&lt;/ref&gt; 

This can be done either directly, 

 sqlite3  places.sqlite .tables

…or within an ''sqlite3'' session.

&lt;syntaxhighlight lang=sh&gt;
$ sqlite3 places.sqlite 
# […]
sqlite&gt; .tables
moz_anno_attributes    moz_favicons           moz_items_annos      
moz_annos              moz_historyvisits      moz_keywords         
moz_bookmarks          moz_hosts              moz_places           
moz_bookmarks_deleted  moz_inputhistory     
sqlite&gt; 
&lt;/syntaxhighlight&gt;

The tables may vary depending on the browser's version.&lt;ref&gt;[http://web.archive.org/web/20190902112616/http://www.forensicswiki.org/wiki/Mozilla_Firefox_3_History_File_Format Mozilla#Database_Tables Firefox 3 History File Format § Database Tables – Forensics Wiki (revision as of September 2011)]&lt;/ref&gt;

A list of columns within a table can be obtained through &lt;code&gt;pragma table_info&lt;/code&gt;, as shown here in this example:

&lt;syntaxhighlight lang=sh&gt;
$ sqlite3 places.sqlite &quot;pragma table_info(moz_places)&quot;
0|id|INTEGER|0||1
1|url|LONGVARCHAR|0||0
2|title|LONGVARCHAR|0||0
3|rev_host|LONGVARCHAR|0||0
4|visit_count|INTEGER|0|0|0
5|hidden|INTEGER|1|0|0
6|typed|INTEGER|1|0|0
7|favicon_id|INTEGER|0||0
8|frecency|INTEGER|1|-1|0
9|last_visit_date|INTEGER|0||0
10|guid|TEXT|0||0

$ sqlite3 places.sqlite &quot;pragma table_info(moz_historyvisits)&quot;
0|id|INTEGER|0||1
1|from_visit|INTEGER|0||0
2|place_id|INTEGER|0||0
3|visit_date|INTEGER|0||0
4|visit_type|INTEGER|0||0
5|session|INTEGER|0||0
&lt;/syntaxhighlight&gt;

Since the default pipe character in the output could interfere with a command line interpreter's URL highlighting, which makes the selection of URLs less convenient, a custom separator character can be specified using &lt;code&gt;.separator&lt;/code&gt; in the &lt;code&gt;~/.sqliterc&lt;/code&gt; file to prevent this. For example, a space character is specified using &lt;code&gt;.separator &quot; &quot;&lt;/code&gt;. Create that file if it does not exist.&lt;ref&gt;[https://database.guide/change-separator-to-comma-in-sqlite-query-results/ Change the Separator to a Comma in SQLite Query Results (2020-04-13)]&lt;/ref&gt;

== Reading the data ==
Tables from the database can be read through the &lt;code&gt;SELECT&lt;/code&gt; operation, with table columns as a dot notation after table names. For example, the following command obtains time stamps and URLs.&lt;ref&gt;{{cite web |url=https://linuxconfig.org/sqlite-linux-tutorial-for-beginners |title=SQLite Linux Tutorial for Beginners |author=Dale Edmons |date=20 September 2021 |access-date=16 February 2022 }}&lt;/ref&gt;

&lt;syntaxhighlight lang=sh&gt;
sqlite3 places.sqlite &quot;SELECT datetime(moz_historyvisits.visit_date/1000000,'unixepoch'), moz_places.url,title FROM moz_places, moz_historyvisits WHERE moz_places.id = moz_historyvisits.place_id&quot; |sort |less -i
&lt;/syntaxhighlight&gt;

An example output line is:
&lt;pre&gt;
2012-04-15 15:56:56|http://www.youtube.com/watch?v=CiuK01bf9X8|Die Pinguine aus Madagascar - Rico Intel Werbung (Operation Super Rico) - YouTube
&lt;/pre&gt;

With custom space separators:
&lt;pre&gt;
2012-04-15 15:56:56 http://www.youtube.com/watch?v=CiuK01bf9X8 Die Pinguine aus Madagascar - Rico Intel Werbung (Operation Super Rico) - YouTube
&lt;/pre&gt;

The &lt;code&gt;datetime(moz_historyvisits.visit_date/1000000,'unixepoch')&lt;/code&gt; part is responsible for converting the Unix epoch time written in the database into a human-readable date and time stamp such as &lt;code&gt;{{#time:Y-m-d H:i:s}}&lt;/code&gt;.

The &lt;code&gt;|sort&lt;/code&gt; pipe makes the entries appear chronologically. This can be desirable for exporting since the order the entries are written in the database is not exactly chronological. Alternatively, &lt;code&gt;ORDER BY visit_date&lt;/code&gt; can be used if natively supported by your ''sqlite'' instance:

&lt;syntaxhighlight lang=sh&gt;
sqlite3 places.sqlite &quot;SELECT datetime(moz_historyvisits.visit_date/1000000,'unixepoch'), moz_places.url,title FROM moz_places, moz_historyvisits WHERE moz_places.id = moz_historyvisits.place_id ORDER BY visit_date&quot; |less -i
&lt;/syntaxhighlight&gt;

After &lt;code&gt;ORDER BY visit_date&lt;/code&gt;, sorting order can be specified using &lt;code&gt;ASC&lt;/code&gt; (ascending) or &lt;code&gt;DESC&lt;/code&gt; (descending).

Finally, the &lt;code&gt;|less&lt;/code&gt; pipe allows directly scrolling through the entries in the terminal software through [[:w:less (Unix)|the &quot;less&quot; utility]] which enables interactive bi-directional scrolling using the arrow keys and &lt;kbd&gt;pgUp&lt;/kbd&gt; and &lt;kbd&gt;pgDn&lt;/kbd&gt; and string searching using &quot;&lt;kbd&gt;/&lt;/kbd&gt;&quot; (slash). The &lt;code&gt;-i&lt;/code&gt; parameter makes string searching case-insensitive. Case-sensitivity can also be toggled within the utility by typing &lt;code&gt;-i&lt;/code&gt;.

Optionally, the history can be exported by redirecting the output into a text file by using &lt;code&gt; &gt;&gt;filename.txt&lt;/code&gt; instead of &lt;code&gt;|less&lt;/code&gt;.&lt;ref&gt;{{cite web |url=https://aurelieherbelot.net/web/read-firefox-history-linux-terminal/ |title=How to read your Firefox history from the terminal |author=Aurelie Herbelot |access-date=2022-02-15 }}&lt;/ref&gt;

This also allows viewing individual time stamps for repeated visits of the same page, since Firefox records a time stamp for each visit, whereas Google Chrome, while recording visit count, overwrites the previous timestamp, meaning only the most recent visit date/time is recorded by Chrome.&lt;ref&gt;{{Cite web|url=https://stackoverflow.com/questions/45644269/chrome-history-possible-to-see-date-of-each-visit-restore-history-to-a-previ|title=Chrome History - possible to see date of each visit / restore history to a previous date?|website=Stack Overflow|access-date=2022-03-04}}&lt;/ref&gt; On Firefox, the former XUL-based extension &quot;Norwell history tools&quot; enabled viewing individual visit times too,&lt;ref&gt;{{Cite web|url=http://web.archive.org/web/20181102045141/https://addons.mozilla.org/en-US/firefox/addon/norwell/|title=Norwell History Tools :: Add-ons for Firefox|date=2018-11-02|website=web.archive.org|access-date=2022-03-04}}&lt;/ref&gt; but XUL extensions are unsupported on Firefox versions 57 &quot;Quantum&quot; (November 2017) onward, and no known successor to the extension has been released since as of 2022.

The form history can be viewed using this SQL line:
&lt;syntaxhighlight lang=sh&gt;
sqlite3 formhistory.sqlite &quot;SELECT datetime(lastUsed/1000000,'unixepoch'),fieldname,value FROM moz_formhistory ORDER BY lastUsed DESC&quot; |less
&lt;/syntaxhighlight&gt;

=== Shortcut function ===
Opening the database file can be facilitated through a shortcut function:
&lt;syntaxhighlight lang=sh&gt;
$ mozhist() { sqlite3 &quot;$1&quot; &quot;SELECT datetime(moz_historyvisits.visit_date/1000000,'unixepoch'), moz_places.url,title FROM moz_places, moz_historyvisits WHERE moz_places.id = moz_historyvisits.place_id ORDER BY visit_date&quot; |less -i; }
$ mozhist places.sqlite
&lt;/syntaxhighlight&gt;

To load this function automatically, add it to the &lt;code&gt;~/.bashrc&lt;/code&gt; file.

=== Raw data ===
To extract the raw data from the database in human-readable form, run &lt;code&gt;sqlite3 places.sqlite .dump |less&lt;/code&gt;. This can be helpful for performing simple title or URL searches (&lt;code&gt;|grep search_string&lt;/code&gt;), but not for dates, since they are not stored in a human-readable format in the database.

== Repairing a corrupted database ==
If Firefox happens to rename the &lt;code&gt;places.sqlite&lt;/code&gt; file to &lt;code&gt;places.sqlite.corrupt&lt;/code&gt;, it means it has detected an error whlist reading the database. To fix it, do the following after choosing the browser's profile folder as working directory:

&lt;syntaxhighlight lang=sh&gt;
# rename to make place
mv places.sqlite places.sqlite.old 
mv places.sqlite.corrupt places.sqlite.corrupt.old

# create dump from corrupted data base
sqlite3 places.sqlite.corrupt.old &quot;.dump&quot; &gt;&gt;dump.sql 

# rebuild database from dump
sqlite3 places.sqlite &quot;.read dump.sql&quot; 
&lt;/syntaxhighlight&gt;

&lt;ref&gt;[https://itectec.com/superuser/how-to-repair-a-corrupted-firefox-places-sqlite-database/ Firefox – How to repair a corrupted Firefox places.sqlite database – iTecTec]&lt;/ref&gt;

This is not guaranteed to fix the problem, but it is worth a try.

== See also ==
* [[Chromium browsing history database]], for Google Chromium-based browsers, most popularly Google Chrome.

== References ==
&lt;references /&gt;

[[Category:Computer Forensics]]
[[Category:SQLite]]</text>
      <sha1>1dp5eq6vtwb0di3s0v4tl2sml37ppv1</sha1>
    </revision>
  </page>
  <page>
    <title>Finding Common Ground</title>
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    <id>282408</id>
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        <username>Dronebogus</username>
        <id>3054149</id>
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      <comment>/* Reality */ a low quality AI image of a man holding a sign? Why?</comment>
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      <text bytes="101210" sha1="77cwi7vlxcnd445ksr3csskufxow0is" xml:space="preserve">— Aligning concepts with reality.
[[File:Layers of Abstraction.jpg|thumb|We engage reality at various layers of abstraction.&lt;ref&gt; This diagram is used as the primary organizing structure for the course. Each ring in the diagram corresponds to a section of the course. Notice that each boundary is blurred. This blurring acknowledges that the various layers interact and no sharp boundary between layers exists. &lt;/ref&gt;]] 
{{TOC right | limit|limit=2}} 
== Introduction ==
Although we all live in the same world and share a single reality, we often seem to be worlds apart when discussing important issues. What is going on? How can we find common ground?&lt;ref&gt;According to [[w:Aumann's_agreement_theorem|Aumann's agreement theorem]], we will be able to find common ground.&lt;/ref&gt;
== Objectives ==
{{100%done}}{{By|lbeaumont}}
The objectives of this course are to:
*Understand the nature of reality.
*Identify the many layers of abstraction through which we encounter reality.
*Navigate through these layers of abstraction.
*Diagnose reasons for conflict during discussions.
*Find common ground.

This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]], the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking curriculum]], and the [[Coming Together]] curriculum.

Use this [[/Daily Practice: Finding Common Ground/|daily practice checklist]] to make finding common ground a habit. 

[https://www.academia.edu/106815969/Finding_Common_Ground Slides based on this course] are available, along with a [https://youtu.be/pQMEDRVwI5U video presentation] of those slides. 

If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont|click here to send me an email]] or leave a comment or question on the [[Talk: Finding Common Ground|discussion page]].

[[File:Finding Common Ground Audio Dialogue.wav|thumb|Finding Common Ground Audio Dialogue]]

== Importance ==
Finding common ground is an important skill because it is useful for resolving the [[w:Social_polarization|social]], [[w:Culture_war|cultural]], [[w:Political_polarization|political]], and [[w:Economic_inequality|economic]] polarization that is prevalent. [[w:Religious_violence|Religious conflicts]], [[w:War|international conflicts]], [[w:Nationalism|nationalism]], [[w:Class_conflict|class struggles]], [[w:Ethnic_conflict|racial tensions]], [[w:Culture_war|culture wars]], ideological conflicts, [[w:Gender_inequality|gender inequality]], [[w:Political_polarization|political polarization]], [[w:Book_censorship|book banning]], [[w:Hate_crime|hate crimes]], [[w:Climate_change_denial|climate change denial]], and [[w:Conspiracy_theory|conspiracy theories]] are a few examples of the conflicts that are dividing people throughout the world.

When we can recognize that [[Facing Facts/Reality is our common ground|reality is our common ground]], we can restore the [[Virtues/Civility|civility]] that is essential to peaceful coexistence. 

== In a Nutshell ==
Here is a brief introduction to the key concepts presented in this course. [[Finding_Common_Ground#Introduction|The diagram]] can help you organize, recall, share, and apply these ideas.
#[[Finding_Common_Ground#Reality|Reality]] exists and we all share in a single objective reality.&lt;ref&gt;{{Cite web|url=https://substack.com/inbox/post/174781036|title=A Minimum Viable Metaphysics|last=Substack|website=substack.com|language=en|access-date=2025-09-30}}&lt;/ref&gt;
#There are several reasons why this fact is difficult for us to grasp and hold onto.
##[[Finding_Common_Ground#Reality|Reality]] is vast, complex, and dynamic. Each of us directly encounters only a tiny slice of reality. We each experience only a glimpse of our vast universe.
##Our direct contact with reality is through our [[Finding_Common_Ground#Perception|perceptions]], which introduce omissions, distortions, and additions.
##Although we use words and other symbols to [[Finding_Common_Ground#Representation|represent]] reality as we perceive it, these symbols are limited, and they only provide approximate representations of our perceptions.
##Because much of what we encounter is [[Embracing Ambiguity|ambiguous]], it invites us to [[Finding_Common_Ground#Interpretation|interpret]] the information to resolve the ambiguity and provide us the comfort of certainty. Many cognitive biases influence our interpretations.
##We love telling and retelling [[Finding_Common_Ground#Narration|stories]]. We easily substitute alluring stories for the complexities and difficulties of reality.
##[[Finding_Common_Ground#Ideology|Ideologies]] substitute a simplified belief system for the complexities of reality. We are easily attracted to these easy to use explanations.
#We can see beyond these illusions and better comprehend reality.
#Reality is our common ground. We can each find that common ground by advancing toward the center of [[Finding_Common_Ground#Introduction|the diagram]] shown above.
#Although these ideas are simply stated, they are difficult to fully grasp and put into practice. Please complete the remainder of this course and use these insights every day.
#We can find common ground.

== Reality ==
For many practical reasons, this course begins with the assumption that [[w:Reality|reality]] exists. Everyday experience provides [[Evaluating Evidence|evidence]] that reality exists. Every time you decide to open the door before passing through the doorway, you are betting that reality exists. If you have lost your keys, then opening the door, or starting your car can become a real problem. If you have difficulty levitating, leaping tall buildings in a single bound, seeing through brick walls, teleporting, or time travelling, perhaps it is because you are encountering constraints imposed by reality.

Despite empirical evidence, people often [[Does objective reality exist?|argue against the existence of reality]]. In these arguments people may cite the [[w:Allegory_of_the_cave|allegory of the cave]], the [[w:Brain_in_a_vat|brain in a vat]], the [[w:Simulation_hypothesis|simulation hypothesis]], the [[w:The_Matrix_(franchise)|Matrix movies]], claims that [[Facing_Facts/Perceptions_are_Personal|perception is reality]], and [[w:Postmodernism|postmodern theories]].

While these are fascinating thought experiments that do deserve some serious philosophical investigation, they don’t provide much help in getting through our daily lives. I bet that reality exists.&lt;ref&gt; Although I am very confident that reality exists, no one can be [[w:Certainty|certain]]. Much of this course demonstrates the value of embracing and exploring [[w:Doubt|doubt]] while deferring certainty. Also, it is often [[Wisdom|wise]] to think in bets. See: {{cite book |last=Duke |first=Annie |author-link=w:Annie_Duke |date= |title=Thinking in Bets: Making Smarter Decisions When You Don't Have All the Facts |publisher=Portfolio |pages= 288 |isbn=978-0735216372}} &lt;/ref&gt;

For the remainder of the course, we proceed with the assumption that reality exists.

Reality is vast, complex, and dynamic. Humans have only investigated a small portion of the universe, and our investigation is incomplete. Our awareness, observations, and perceptions of reality are neither complete nor accurate representations of reality. We do not observe the many [[w:Cosmic_ray|cosmic rays]] passing through us each second, the atoms that make up the materials we encounter, billions of galaxies beyond the limits of our vision, the ultrasonic chirps used by bats to navigate, viruses, DNA, antibodies, [[w:Greenhouse_gas |greenhouse gasses]], [[w:Particulates|particulate contaminants]], and much more of reality as it is.

This course makes the further assumption that [[Knowing_How_You_Know/One_World|we all live in the same universe]]&lt;ref&gt;This assumption does not conflict with [[w:Many-worlds_interpretation|many-worlds interpretations]] of quantum mechanics. &lt;/ref&gt;. All life forms discovered so far live together on our single planet, circling our sun, in our humble place in the universe. The universe is vast, yet it is all one world, and we all live together on this one planet we call Earth. All that we know of and all that we have ever experienced follow the same laws of physics. Remarkably, the entire world as we know it has emerged from a few [[w:Standard_Model|fundamental building blocks]].

Because we all live in the same universe, our reliable understanding of that universe must eventually converge toward one coherent description. Each phenomenon we observe must fit into a single coherent and integrated description of our universe. Either the description must evolve to accommodate each new observation, or our understanding of that observation must be interpreted consistently with that unified representation.

Building on these assumptions, it follows that [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]], even though any number of [[Exploring Worldviews|worldviews]] is possible. It is likely that each of us hold a worldview that is somewhat different from others. Of all the possible worldviews, one worldview is especially important. That is the worldview that corresponds to reality as closely as our best current understanding of reality allows. Because reality exists, we can [[Evaluating_Evidence|examine reality]], and we can [[Exploring_Worldviews/Aligning_worldviews|align our worldview]] with reality. Because we all live in one world, reality is our common ground.
=== Assignment ===
#Complete the Wikiversity course on [[Facing Facts]].
#Read the essay [[Facing_Facts/Perceptions_are_Personal|Perceptions are personal]].
#Investigate further any of the arguments that appeal to you, listed above, that reality does not exist.
#If you can accept the assumption that reality exists, please proceed with the remainder of this assignment.
#Estimate what fraction of reality you are familiar with.
##Read the essay, [[Virtues/Humility/Being_99.9%_Ignorant|Being 99.9% Ignorant]].
##How many countries are there in the world? How many have you visited?
##Study modern research on the [[w:Galaxy#Modern_research|number of galaxies in the universe]]. How many have you visited?
##Scan this list of [[w:Lists_of_unsolved_problems|lists of unsolved problems]]. What fraction of reality has been investigated by humans?
##Can you accept the premise that reality exists far beyond our perceptions, investigations, and conceptions of it?
#Read the essay [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]].
#Read the essay [[Knowing_How_You_Know/One_World|One World]].
#Read the essay [[Facing_Facts/Reality_is_our_common_ground|Reality is our common ground]].
#Read the essay [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]].
#Optionally read the essay [[Beyond_Theism/What_there_is|What there is]]. Consider writing your own such essay.  
#Can you accept the premise that reality exists, we live in one world, we share one reality, and that reality  is our common ground? If so, please proceed with the remainder of this course.

== Perception ==
[[w:Perception|Perception]] transforms sensory information originating with some material object, known as the ''target'' or the ''[[w:Distal|distal]] stimulus'' into mental representations known as [[w:Perception#Process_and_terminology|percepts]], which do not exist elsewhere in the world. 

[[w:Perception|Perception]]—the process of extracting information from energy that impinges on our [[w:Sense|sensory organs]]—is not straightforward. There is more to perception than meets the eye.&lt;ref&gt;{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }}Page 13 of 162.&lt;/ref&gt;

Indirect theories of perception describe it as a constructive process that involves inference, learning, and experience.&lt;ref&gt;{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }}Page 13 of 162.&lt;/ref&gt; 

While it may seem reasonable that an ideal perception system would duplicate the observed target object exactly in the mental percept, this is not what happens. One theory hypothesizes that the purpose of perception is to allow organisms to locate and use [[w:Affordance|affordances]]—what the environment offers the individual—for practical benefit.

Our most direct encounters with reality are through direct sensory observations and awareness. However, our perceptions are not exact replicas of the distal objects being perceived. Omissions, distortions, and additions take place. Fortunately, we can augment our direct perceptions to obtain a more accurate understanding of our world.

=== Omissions ===
Our direct perception accounts for only a tiny fraction of reality.

We can only directly perceive stimuli that our senses encounter, and these senses experience only a small fraction of reality. We do not see what is behind us, the places we do not visit, or stimuli outside the range of our sensory systems. Attention is selective, so we are likely to perceive only what our attention is drawn to, remaining unaware of the rest. When our attention is captured by moving objects, shiny objects, or unexpected occurrences, we may be distracted from sensing other information in our environment.
[[File:Directing_Attention.jpg|300px|thumb|Our perceptions focus on what we direct our attention toward.]]
Although reality is our common ground, each of us has unique experiences of its vastness. Refer to the diagram on the right. Reality is vast, complex, and dynamic, and the range of experiences any of us has is a tiny fraction of reality as it is. Furthermore, within our range of experience, our attention is limited. Our attention concentrates awareness on some phenomena to the exclusion of other stimuli.&lt;ref&gt;The idea of representing attention as a vertical line sampling reality is presented in {{cite book|title=Liminal Thinking: Create the Change You Want by Changing the Way You Think  |last=Gray|first=Dave|date=September 14, 2016|publisher=Two Waves Books|isbn=978-1933820460|pages=184|author-link=}}&lt;/ref&gt; Because our concentration is selective and focuses on discrete information, whether subjectively or objectively, we exclude other stimuli present even within our range of experience.

Our perceptions focus on what we direct our attention toward.

The spectrum of [[w:Visible_spectrum|light visible to humans]] is only a tiny fraction of the [[w:Electromagnetic_spectrum|electromagnetic spectrum]]. Radio waves, microwave radiation, infrared radiation, visible light, ultraviolet radiation, x-rays, and gamma radiation are all electromagnetic waves of various wavelengths. These various forms of radiation span wavelengths ranging from as small as 10&lt;sup&gt;-11&lt;/sup&gt; meters (0.1 Å) to as long as 1,000 meters, however only light in the range of 400 [[w:Nanometre|nm]] – 700 nm (4x10&lt;sup&gt;-7&lt;/sup&gt; to 7x10&lt;sup&gt;-7&lt;/sup&gt; meters) is visible to humans. We have no way to directly sense radio waves, infrared and ultraviolet light, x-rays, and other forms of radiation. Humans are also unable to directly sense magnetic fields.

The [[w:Hearing_range|range of human hearing]] is approximately 20 to 20,000 [[w:Hertz|Hz]], while dogs can hear sounds frequencies as high as 45 kHz and bats can hear frequencies as high as 200 kHz. Elephants can hear sounds at 14–16 Hz, while some whales can hear [[w:Infrasound|infrasound]] as low as 7 Hz.

It is estimated that dogs, in general, have an [[w:Sense_of_smell#Variability_amongst_vertebrates|olfactory sense]] (sense of smell) approximately ten thousand to a hundred thousand times more acute than a human's.

Also, we are often [[Exploring_Worldviews/What_Fish_Don’t_See|unaware of what could be most obvious]]. 

Not only does our direct perception account for only a tiny fraction of reality, much of what we do perceive is distorted.

=== Distortions: ===
Distortions of the [[w:Sense|senses]] are called [[w:Illusion|illusions]]. Although illusions distort our perception of reality, they are generally shared by most people. Illusions may occur with any of the human senses, but visual illusions (optical illusions) are the best-known and understood.

[[File:Checker shadow illusion.svg|thumb|The squares marked A and B are the same shade of gray.]]
In the [[w:Checker_shadow_illusion|checker shadow illusion]], shown here, two identically colored squares labeled “A” and “B” appear to be very different shades of gray. Reality as we perceive it may not be reality as it is.

[[Image:Fraser spiral.svg|thumb|right|225px|Fraser spiral illusion]]
In the Fraser spiral illusion, shown here, the overlapping black arc segments appear to form a spiral; however, the arcs are a series of concentric circles.

[[File:Duck-Rabbit illusion.jpg|thumb| What do you see? Are you sure?]]
In addition to illusions, many [[w:Ambiguous_image|images are ambiguous]]. As an example, please observe, and then interpret the image shown here.

What do you see? What else do you see? How certain are you of your conclusion? What do you have at stake in defending your interpretation?

Most people will see either a rabbit or a duck. If you see a rabbit, look again, and try to see a duck. Similarly, if you see a duck, look again, and try to see a rabbit.

This image, called the [[w:Rabbit-duck_illusion|rabbit-duck illusion]] is inherently [[w:Ambiguity|ambiguous]]. Someone who interprets it as representing a duck has as valid a claim to accuracy as someone who interprets this as representing a rabbit. In fact, it is neither and both. It is an ambiguous image open to at least two different yet equally valid interpretations. Similarly, a [[w:Is_the_glass_half_empty_or_half_full?|glass that is half empty]] is also half full.

In addition to optical illusions, we are susceptible to illusions that distort each of our senses. These include [[w:Auditory_illusion|auditory]], [[w:Tactile_illusion|tactile]], [[w:Time_perception#Types_of_temporal_illusions|temporal]], and [[w:Illusion#Intersensory|intersensory illusions]].

[[w:Magic_(illusion)|Magicians]] are performing artists who use illusions to entertain us. [[w:Charlatan|Charlatans]], including fakes, [[w:Mysticism|mystics]], [[w:Counterfeit|counterfeiters]],  [[w:Quackery|quacks]], [[w:Conspiracy_theory|conspiracy theorists]], and [[w:Confidence_trick|con artists]], use illusions to deceive, cheat, and swindle us. [[w:Ideology|Ideologs]] exploit illusions to promote their cause, [[w:Special_effect|special effects]] are often used in films and other artistic media to entertain us, and recently [[w:Deepfake|deepfakes]] are used to deceive us. 

In addition to our perceptions being distorted, we also sometimes perceive things which are not there. 

=== Additions: ===
Our perception systems create [[w:Perception#Process_and_terminology|percepts]] that do not otherwise exist in the world. Examples include [[w:Color_vision|color perception]], [[w:Illusory_contours|illusory contours]] and other patterns studied within [[w:Gestalt_psychology|Gestalt psychology]], [[w:Apophenia|illusory pattern recognition]], pain, phantom limbs, [[w:Hallucination|hallucinations]], and distortions that occur during various [[w:Altered_state_of_consciousness|altered states of consciousness]]. Our conceptions also influence our perceptions.

Humans [[w:Perception|perceive]] the [[w:Red|color red]] when we look at light with a wavelength between approximately 625 and 740 [[w:Nanometre|nanometers]]. Note that although electromagnetic radiation with a wavelength of approximately 700nm does exist in objective reality independent of the direct perception of any human, the percept we call red only exists in  our minds as a construct of the perceptual system. This is a consequence of our [[w:Trichromacy|trichromatic color vision]] system, including our visual cortex and [[w:Color_vision#Color_in_the_human_brain|other brain regions]]. Because color only exists as a feature constructed by the visual perception of an observer it is a [[w:Color_vision#Subjectivity_of_color_perception|subjective experience]].  When I see red, I have a subjective experience called redness. However, philosopher John Locke proposed a thought experiment, called the [[w:Inverted_spectrum|inverted spectrum]], where we imagine two people sharing their color vocabulary and discriminations, although the colors one sees—one's [[w:Qualia|qualia]]—are systematically different from the colors the other person sees. For example, perhaps the color we call red creates within me the subjective experience that the color we call green generates within you. We do not know if the subjective experience I call redness is like the subjective experience you have when seeing red.&lt;ref&gt;Siegel, Susanna, &quot;[https://plato.stanford.edu/archives/fall2021/entries/perception-contents The Contents of Perception]&quot;, The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), Edward N. Zalta (ed.), Section 4.1.&lt;/ref&gt;  The nature of redness is further explored in a philosophical thought experiment known as the [[w:Knowledge_argument|knowledge argument]], or Mary’s room.

[[Image:Kanizsa triangle.svg|thumb|right|''Kanizsa's triangle'': Do you see a white triangle?]]

This image, called [[w:Illusory_contours|Kanizsa’s triangle]], gives the impression of a bright white triangle, defined by a sharp illusory contour, occluding three black circles and a black-outlined triangle. Even knowing the bright white triangle does not exist, it is reliably constructed by our visual perception system. This is one example of many [[w:Illusory_contours|illusory contours]] that evoke the perception of an edge without a shading or color change across that edge. We see an edge where none exists. [[w:Gestalt_psychology|Gestalt psychology]] studies many such images where it appears that “The whole is more than the sum of its parts.”

[[w:Apophenia|Apophenia]] is the tendency to perceive meaningful connections between unrelated things. For example Gamblers may imagine that they see patterns in the numbers that appear in lotteries, card games, or roulette wheels, where no such patterns exist.

[[File:Martian face viking cropped.jpg|thumb|Satellite photograph of a [[w:mesa | mesa]] in the [[w:Cydonia (Mars)|Cydonia region of Mars]], often called the [[w:Cydonia (Mars)#&quot;Face on Mars&quot;|&quot;Face on Mars&quot;]] and cited as evidence of [[w:Extraterrestrial life|extraterrestrial]] habitation.]]

One form, called [[w:Pareidolia|pareidolia]], is the tendency for perception to impose a meaningful interpretation on a nebulous stimulus, usually visual, so that one sees an object, pattern, or meaning where there is none. People may mistakenly interpret an object, shape, or configuration with perceived &quot;face-like&quot; features as being a face. Many people claim to see the [[w:Man_in_the_Moon|man in the moon]]. In this satellite photograph of a [[w:mesa | mesa]] in the [[w:Cydonia (Mars)|Cydonia region of Mars]], often called the [[w:Cydonia (Mars)#&quot;Face on Mars&quot;|&quot;Face on Mars&quot;]] has been cited as evidence of [[w:Extraterrestrial life|extraterrestrial]] habitation.

As another example, what we feel as [[w:Pain|pain]] is our perception of tissue damage. Consider our perception of a [[w:Toothache|toothache]]. The damaged tooth stimulates nerve impulses in our [[w:Somatosensory_system|somatosensory nervous system]]. The brain perceives various nerve impulses transmitted through these neural structures as the unpleasant sensory and emotional experience we call pain. Notice that the brain receives only a pattern of nerve impulses and then constructs the perception of pain from those nerve impulses. Although the damage is in the tooth the perception of pain is constructed in the brain.

A [[w:Phantom_limb|phantom limb]] is the sensation that an amputated or missing limb is still attached. Approximately 80 to 100% of individuals with an amputation experience sensations in their amputated limb. In phantom limb syndrome, there is sensory input indicating pain from a part of the body that no longer exists. This phenomenon is still not fully understood, but it is hypothesized that it is caused by activation of the [[w:Somatosensory_cortex|somatosensory cortex]]. The perception of the absent limb is constructed by our nervous system.

As another example, [[w:Tinnitus|tinnitus]] is the perception of sound when no corresponding external sound is present.

Our perceptions are often distorted when we are experiencing [[w:Altered_state_of_consciousness|altered states of consciousness]]. This may occur from the influences of drugs—especially [[w:Psychoactive_drug|psychoactive drugs]] including [[w:Hallucinogen|hallucinogens]] and [[w:Alcohol_(drug)|alcohol]]—fatigue, disease, [[w:Hypoxia_(medical)|hypoxia]] (as can occur from [[w:Hypoventilation|hypoventilation]] and other [[w:Breathing|breath]] control practices), [[w:Hallucination|hallucinations]], [[w:Delirium|delirium]], and various mental illnesses. 

Our conceptions often influence our perceptions. Try this simple quiz:
&lt;blockquote&gt;
What does F-O-L-K spell? (Please say the word out loud.)&lt;br&gt;
What is the white of an egg called?  (Please say the word out loud.)&lt;br&gt;
(What color is the white of an egg?)
&lt;/blockquote&gt;
If you said “yolk”, as many people do, you were influenced by psychological priming. [[w:Priming_(psychology)|Priming]] is a phenomenon whereby exposure to one stimulus influences a response to a subsequent stimulus, without conscious guidance or intention. For example, in experiments the word ''nurse'' is recognized more quickly following the word ''doctor'' than following the word ''bread''.

We may be unaware of priming effects that arise [[w:Priming_(psychology)#In_daily_life|in our daily lives]]. In one study subjects were implicitly primed with words related to the stereotype of elderly people (example: Florida, forgetful, wrinkle). While the words did not explicitly mention speed or slowness, those who were primed with these words walked more slowly upon exiting the testing booth than those who were primed with neutral stimuli.

There is more to perception than meets the eye!

=== Assisted Perception—Compensating for the quirks ===
Fortunately, there are many methods we can use to overcome and compensate for the deficiencies, distortions, quirks, and other characteristics of our direct perception systems. Microscopes, telescopes, [[w:Oscilloscope|oscilloscopes]], many other [[w:Scientific_instrument|scientific instruments]], recording devices, and [[w:Technical_standard|reference standards]] allow us to extend the reach, scope, and accuracy of our observations. Multiple observers, vantage points, perspectives, and viewpoints allow us to gain a more complete and reliable examination of reality when we share and integrate information. 

Applying the principle of [[w:Consilience|consilience]] and [[Thinking Scientifically|thinking scientifically]] increase the reliability of our observations and can help us see beyond illusions. 

==== Assignment ====
#Observe the moon in the sky some evening.
#Estimate the size (diameter) of the moon.
#Estimate the distance the moon is from you.
#Research the [[w:Moon|accepted values of these numbers]] and compare them to your direct observations. How closely does the perceived distance and size compare to the accepted values?
#Optionally repeat this for the sun, bright stars, and other distant terrestrial and celestial objects. 

=== Perceptions are Personal ===
We often hear that “perception is reality” and that “everything is relative”, despite knowing that a shared reality exists, and reality is our common ground.

Perceptions are vivid. Seeing things from our own point of view is always easier, and first-hand experiences seem more real than understanding another's point of view can ever be. Our eyes, nose, taste buds, tactile sensors, and ears connect directly only to our brain. Only you experience first-hand the direct sensory input of the world; you, your self, is the observer. This raw sensory input is interpreted and gains meaning through your unique perceptions and past experiences. Furthermore, contemplation, desire, intent, pain, introspection, consciousness, and reflection are all private and solitary. This unique first-person experience creates a fundamental asymmetry that contributes to many of the other asymmetries that govern social interactions. It also contributes to the asymmetric character of [[Coping with Ego|egotism]], narcissism, selfishness, greed, and the magnitude gap. We judge others based on behavior and we judge ourselves based on intent. Your own point of view, the way you see things, is unique. The [[Living the Golden Rule|golden rule]] and our empathy struggle to overcome this fundamental imbalance.

It is often a [[w:Problem_of_induction|mistake to generalize]] our personal perceptions beyond our own experiences. Standing in a meadow we see a flat earth, yet sunrise, time zones, global travel, earth satellites, GPS navigation systems, images from space, and travel to the moon all assure us the [[w:Spherical_Earth|earth is nearly spherical]]. Reality is vast, complex, and dynamic, and our perceptions are only a tiny glimpse of all there is to know about reality. Only a [[Global Perspective|global perspective]] brings us uncensored reality.

Reality exists and provides us with [[Facing_Facts|matters of fact]]. The [[Knowing_How_You_Know/Height_of_the_Eiffel_Tower|Eiffel tower is 300 meters tall]]. You may perceive that as too short; others may perceive that as too tall, and many perceive that as simply beautiful.
 
See beyond the illusion that what you see is all there is. Perceptions are personal, but [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]]. Our unique first-person viewpoint creates a powerful asymmetry that requires deliberate effort to see beyond. Transcend your personal perceptions, investigate the vastness and complexity of reality from a [[Global Perspective|global perspective]], and embrace reality as our common ground.
==== Assignment ====
#Read the essay [[Facing_Facts/Perceptions_are_Personal|Perceptions are Personal]].
#Reflect on occasions when you projected your own perceptions onto others or beyond your own experience.
#Recognize the limited scope and likely deficiencies of your perceptions.

== Representation ==
At least until we become proficient at [[w:Telepathy|mental telepathy]], we need some other way to represent our thoughts and to communicate our thoughts to others.

[[w:Word|Words]] are one type of [[w:Symbol|symbol]] used to represent reality. Other symbols might include [[w:Vocable|sounds]], [[w:List_of_gestures|gestures]], [[w:Facial_expression|facial expressions]], [[w:Icon_(computing)|icons]], [[w:Logo|logos]], [[w:Drawing|drawings]], [[w:Model|models]], [[w:Analogy|analogies]], [[w:Metaphor|metaphors]], and other representations. The mapping from words to perceptions to reality [[w:Polysemy|is often imprecise]]. We may describe a certain color simply as ''red'', without clarifying what [[w:Shades_of_red|shade of red]] we are describing. If I tell you I am sitting in a chair, you might imagine a rocking chair, a desk chair, or a recliner. [[w:Syntactic_ambiguity|Syntactic ambiguity]] often adds to the confusion.

Do not mistake the symbol for reality. The [[w:Noumenon|noumenal]] world exists independent of our senses and may differ from the [[w:Noumenon|phenomenal]] world we perceive and the symbols we use to represent it. 

Language and other symbols can be constraining, ambiguous, refer to social constructs, prejudiced, based on artificial boundaries, or focus on convenient representation rather than reality. These features, distortions, and inaccuracies are explored further below.

=== Constraining Language ===
[[File:Clip.jpg| thumb|right|What can you do with this wire form? What would you call it?]]
Consider the image on the right. What can we use it for? This wire form is commonly called a [[w:Paper_clip|paper clip]] and is typically used to clip together sheets of paper. However, [[Thinking_Tools#Lateral_Thinking|lateral thinkers]] have identified at least one [[w:Paper_clip#Other_uses|hundred alternative]] uses&lt;ref&gt;100 Uses for Paperclips, See: https://leoniehallatinnovationiq.wordpress.com/2012/11/21/100-uses-for-paperclips/ &lt;/ref&gt;  for this object. Observing the object, thinking expansively and creatively about what it might be, how it can be used, or what it might do, can identify many possibilities. This unclassified reality invites many potential uses. Once the object is named, an interpretation is imposed, and the object becomes restricted.

Naming the object assigns the object to one category, as it excludes others. 

This table describes possibilities before and after it is interpreted, assigned to a category, and named.

{| class=&quot;wikitable&quot;
|+ Categorization Constrains Imagination
|-
! Encounter with reality: !! Interpretation and Representation:
|-
|
*An experience
*Awareness
*Curious
*Exploration
*Investigation
*Witness
*Imagine
*Possibilities
*Potential
*Opportunities
*An affordance
*Lateral Thinking
*No (one) thing
*The territory
*Could be …
||
*Analogy
*Categories
*Labels
*Restrictions
*Evaluation
*Judgement
*Purpose
*Specifics
*''This'' map
*''This'' thing
**A paperclip
**A lock pick
**A hook
**DVD drive opener
**Earrings
|}
	
Language and other symbols are ambiguous, persuasive, and subtle, and can be used heroically, precisely, expansively, restrictively, deceptively, and manipulatively. Here are some examples.

=== Ambiguous Language ===
[[w:Polysemy|'''Polysemy''']] is the capacity for a sign (e.g., a symbol, a word, or a phrase) to have multiple related meanings. A word can have several [[w:Polysemy|word senses]]. As an example, the word ''bank'' can have at least these 7 distinct meanings:
#a financial institution
#the physical building where a financial institution offers services
#to deposit money or have an account in a bank (e.g., &quot;I bank at the local credit union&quot;)
#[[w:Bank_(geography)|a steep slope]] (as of a hill or the rising ground bordering water)
#[[w:Banked_turn#Banked_turn_in_aeronautics|to incline an airplane]] laterally
#a supply of something held in reserve: such as &quot;banking&quot; brownie points
#a synonym for 'rely upon' (e.g. &quot;I'm your friend, you can bank on me&quot;).

[[w:Syntactic_ambiguity|'''Syntactic ambiguity''']] is language where a [[w:Sentence_(linguistics)|sentence]] may be interpreted in more than one way due to ambiguous [[w:Syntax|sentence structure]]. For example, the sentence ''John saw the man on the mountain with a telescope'' can have these various interpretations. 
*John, using a telescope, saw a man on a mountain.
*John saw a man on a mountain which had a telescope on it.
*John saw a man on a mountain who had a telescope.
*John, on a mountain and using a telescope, saw a man.
*John, on a mountain, saw a man who had a telescope.

Combining polysemy with syntactic ambiguity results in multiplying the ambiguity. For example, Gerald Weinberg identifies dozens of interpretations of the sentence “Mary had a little lamb”&lt;ref&gt; {{cite book |last1=Gause  |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is  |publisher=Dorset House Publishing Company |pages=176 |isbn=978-0932633163 |author-link=w:Gerald_Weinberg}} @128 of 255.&lt;/ref&gt;  and then goes on to suggest 20 more techniques for identifying ambiguity in sentences.

[[Exploring_Social_Constructs#Reifications|'''Reifications''']] are a class of nouns that do not refer to real objects but rather to abstract concepts. Examples include justice, freedom, liberty, equality, rights, duty, responsibility, truth, fairness, and citizen and extend to include concepts such as race, land ownership, owning coal, money, debt, contracts, and even agreement. These labels cannot be resolved to [[w:Brute_fact#Searle|brute facts]] but are often treated as if they do. This results in a [[w:Reification_(fallacy)|reification fallacy]]. A [[w:Map–territory_relation#&quot;A_map_is_not_the_territory&quot;|map is not the territory]], an [[w:The_Treachery_of_Images|image of a pipe is not a pipe]], and there is little or no agreement on what “[[w:justice|justice]]” is. 

Simple phrases such as “we seek justice”, “justice was served”, “[[w:Pledge_of_Allegiance|liberty and justice for all]]”, “[[w:Preamble_to_the_United_States_Constitution|establish justice]]”, and “ours is a nation of laws” become very ambiguous, complex, and controversial when we recognize the wide range of meanings attributed to the abstract concept of “justice”.

The concept of [[w:justice|justice]] differs in every culture. Early theories of justice were set out by the Ancient Greek philosophers Plato in his work [[w:Republic_(Plato)|The Republic]], and Aristotle in his [[w:Nicomachean_Ethics|Nicomachean Ethics]]. Throughout history various theories have been established. Advocates of [[w:Divine_command_theory|divine command theory]] argue that justice issues from God. In the 1600s, theorists like [[w:John_Locke|John Locke]] argued for the theory of [[w:Natural_law|natural law]]. Thinkers in the [[w:Social_contract|social contract]] tradition argued that justice is derived from the mutual agreement of everyone concerned. In the 1800s, [[w:Utilitarian|utilitarian]] thinkers including [[w:John_Stuart_Mill|John Stuart Mill]] argued that justice is what has the best consequences. Theories of [[w:Distributive_justice|distributive justice]] concern what is distributed, between whom assets or liabilities are to be distributed, and what is the ''proper'' distribution. [[w:Egalitarianism|Egalitarians]] argued that justice can only exist within the coordinates of equality. [[w:John_Rawls|John Rawls]] used a [[w:Social_contract|social contract]] argument to show that justice, and especially distributive justice, is a form of [[Understanding Fairness|fairness]]. Property rights theorists (like [[w:Robert_Nozick|Robert Nozick]]) also take a consequentialist view of distributive justice and argue that property rights-based justice maximizes the overall wealth of an economic system. Theories of retributive justice are concerned with [[w:Punishment|punishment]] for wrongdoing. [[w:Restorative_justice|Restorative justice]] (also sometimes called &quot;reparative justice&quot;) is an approach to justice that focuses on the needs of victims and offenders.

There are many [[w:Wikipedia:Manual_of_Style/Words_to_watch|words to avoid]] when trying to be objective and precise. 
===== Assignment =====
#Choose a speech or other text to use for this assignment. This may be chosen from this [[w:List_of_speeches|list of speeches]], the Wikisource [[s:Portal:Speeches|speeches portal]], [[w:List_of_amendments_to_the_United_States_Constitution|amendments to the United States Constitution]], the [[w:Universal_Declaration_of_Human_Rights|Universal Declaration of Human Rights]], [[w:List_of_landmark_court_decisions_in_the_United_States|landmark court decisions]], or any other source.
#Identify instances of ambiguous language in the chosen text.
#For at least three instances of ambiguous language, list various meanings that can be reasonably implied by the language used. For example, the ambiguous phrase “in the future” could mean seconds, hours, days, weeks, months, years, centuries, or eons from now. 
#Suggest more precise and objective language that could be substituted for the language used. 
#Suggest an [[w:Operational_definition|operational definition]] to replace or clarify the ambiguous language. 
==== Social Constructs ====
When we endow [[w:Brute_fact#Searle|''brute facts'']] with additional status, we create [[Exploring_Social_Constructs|''social constructs'']].

Social constructs rely on collective human agreement, or at least acceptance of the imposition of the new status Y on the brute fact X.

Social constructs include games such as soccer, baseball, and chess. Bureaucracies including clubs, organizations, and corporations are social constructs. Titles such as chairman, president, king, and pope, along with governments, financial instruments, property ownership agreements, and religions are social constructs.

Social constructs are [[Exploring_Social_Constructs#Ambiguity|ambiguous]], sometimes fragile, and often require [[Exploring_Social_Constructs#Referees|referees]] of some form. The agreements used to form the social constructs may be obsolete or challenged. Social constructs can be [[Exploring_Social_Constructs#Mismatches|mismatched]] to relevant brute facts.

Because social constructs are so common and so prominent, we can easily mistake them for brute facts. This is an error. Mistaking social constructs for brute facts introduces several layers of abstraction and creates distortions that distance us from realty. 

As a result of ambiguity and defective agreements many social constructs are poorly aligned with the brute facts they are based on, and many mismatches occur. These cause friction in our society and can contribute to [[Grand challenges|many challenges]] we face. By observing the mismatch of social constructs to brute facts and informed consent, we can begin to troubleshoot and improve the collection of social constructs that create our culture and institutions.

The bad news is that we face many issues resulting from social constructs misaligned with brute facts or based on defective agreements. The good news is that because social constructs are human constructs, we can work to improve them.
===== Assignment =====
#Choose a speech or other text to use for this assignment. This may be chosen from this [[w:List_of_speeches|list of speeches]], the Wikisource [[s:Portal:Speeches|speeches portal]], [[w:List_of_amendments_to_the_United_States_Constitution|amendments to the United States Constitution]], the [[w:Universal_Declaration_of_Human_Rights|Universal Declaration of Human Rights]], [[w:List_of_landmark_court_decisions_in_the_United_States|landmark court decisions]], or any other source.
#Identify instances of ''social constructs'' that appear in the chosen text.
#For at least three instances of the social constructs identified, list various meanings that can be reasonably implied by the language used. For example, the [[w:Second_Amendment_to_the_United_States_Constitution|second amendment]] uses the term “[[w:Militia|Militia]]”.  This could mean military unit, police units, paramilitary units, security guards, or individuals. 
#Suggest more precise and objective language that could be substituted for the language used. 
#Suggest an [[w:Operational_definition|operational definition]] to replace or clarify the ambiguous language. 

==== Prejudiced Language ====
[[w:Loaded_language|Loaded language]] is [[w:Rhetoric|rhetoric]] used to influence an audience by using words and phrases with strong [[w:Connotations|connotations]]. This type of language is unclear (vague) and can be used to [[w:Pathos|invoke an emotional response]] or exploit [[w:Stereotypes|stereotypes]].  Loaded words and phrases have significant emotional implications and involve strongly positive or negative reactions beyond their [[w:Literal_meaning|literal meaning]]. These may include [[w:Racism|racists]] or [[w:Sexism|sexist]] words, [[w:List_of_ethnic_slurs|ethnic slurs]], [[w:List_of_religious_slurs|religious slurs]], and other [[w:Lists_of_pejorative_terms_for_people|pejorative terms]]. 

[[w:Wikipedia:Manual_of_Style/Words_to_watch#Contentious_labels|Examples of contentious labels include]]: cult, racist, perverted, sexist, homophobic, transphobic, misogynistic, sect, fundamentalist, heretic, extremist, denialist, terrorist, freedom fighter, bigot, myth, neo-Nazi, -gate, pseudo-, controversial, and others.
==== Beware of Boundaries ====
[[File:Tannin heap.jpeg|thumb|The [[w:Sorites_paradox|sorites paradox]]: If a heap is reduced by a single grain at a time, at what exact point does it cease to be considered a heap?]]

The [[w:Sorites_paradox|sorites paradox]] poses the question, if removing one grain from a heap of sand leaves it a heap, then one grain of sand is also a heap. When does a heap of sand transform into a few grains of sand that are no longer a heap? This paradox illustrates that the concept of ''heap'' is ambiguous. Also, the boundary between a heap and some smaller collection is also ambiguous. 

In conventional language, logic, mathematics, and decision-making, we generally regard [[Natural_Inclusion/Boundaries|boundaries]] as discrete or definitive limits or borders, which permanently and absolutely divide one thing or locality apart from other things or localities. For such definitive limits to exist, however, they would have to be so sharp as to have no thickness. By contrast, even when viewed from afar and over short durations, natural boundaries often appear diffuse, mobile, and impermanent, defying such precise, abstract definition. Naturally occurring boundaries are inherently ambiguous. Artificial boundaries are often sharply defined. This mismatch between how boundaries occur naturally and how we chose to represent them can lead to several problems. 

Racial classifications create problems because they impose sharply defined boundaries where no natural boundary exists. [[w:Race_(human_categorization)|Racial classifications]] are [[Exploring_Social_Constructs|socially constructed]]. While partially based on physical similarities within groups, race does not have an inherent physical or biological meaning. Therefore, race assignment is inherently ambiguous. None-the less, laws prevail that impose harsh burdens based on racial classification. These laws define sharp boundaries to separate one race from another. Simplistic resolutions of racial ambiguity are the [[w:One-drop_rule|one-drop rules]] that asserted that any person with even one ancestor of black ancestry ('one drop' of 'black blood') is considered black. Attempts to define [[w:Native_American_identity_in_the_United_States#Blood_quantum|Native American identity in the United States]] encounters similar difficulties.

==== The map is not the territory. ==== 
As soon as we label an object to represent it, use a [[w:Mental_model|mental model]], invoke an [[w:Analogy|analogy]], use a [[w:Metaphor|metaphor]], substitute a representation for a thought, idea, or object, or substitute an interpretation for some set of observations we substitute [[w:Map–territory_relation|the map for the territory]].

Our brain learns a model of the world. Intelligence is tied to the model the brain creates. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}@22 of 384&lt;/ref&gt;  The brain learns its model of the world by observing how its inputs change over time. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}} @66 of 384&lt;/ref&gt; The neocortex learns a predictive model of the world. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}@75 of 384&lt;/ref&gt;

We rely on the map our brain creates to navigate the world we live in. This works best when our mental maps correspond accurately to the real world. We navigate through life guided by the mental representations that form in our brains. If you are now sitting on a chair in a room, and you want to leave that room, you rely on your mental image of the chair, your location in the room, and the location of the door. If this mental model is wrong, you will have difficulty getting out of the chair, walking across the room, opening the door, and walking through.

We know [[w:Map–territory relation|the map is not the territory]], but merely some simplified and often distorted representation of the corresponding territory—reality as it is. Whenever we rely on the map rather than the territory for forming beliefs, deciding, or planning actions, we risk misrepresenting the territory and embracing a distorted view of reality.

George Box reminds us that “[[w:All_models_are_wrong|All models are wrong]], some are useful”.

We depart from reality the moment we move from [[w:Perception|''perception'']] to [[w:Concept|''conception'']] from [[w:Observation|''observation'']] to [[w:Interpretation_(philosophy)|''interpretation'']].  In his painting [[w:The_Treachery_of_Images|''The Treachery of Images'']], [[w:René_Magritte|René Magritte]] reminds us that an image of a pipe is a representation of the pipe and not the pipe itself.

It is difficult to be precise and neutral in our language. Language provides many opportunities to be vague, ambiguous, [[w:Bullshit|nonsensical]], prejudicial, emotional, laudatory, persuasive, kind, cruel, [[w:Buzzword|vacuous]], or [[w:Weasel_word|equivocal]]. Because [[w:Rhetoric|rhetoric]] is the art of persuasion, be aware that it can draw you toward a conclusion and influence your beliefs using only baseless arguments and emotional manipulation. 

=== Assignment ===
#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of language, perception, or reality.
#[[Practicing Dialogue|Practice dialogue]] rather than debate or argumentation. Be candid. [[Living_Wisely/Advance_no_falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course.
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[Evaluating_Evidence|research them]].
##[[Seeking True Beliefs|Seek true beliefs]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Interpretation ==
People [[w:Interpretation_(philosophy)|interpret]] observations to resolve ambiguity, increase certainty, and to account for observations within some familiar model or analogy. The role of interpretation is most evident when the items being interpreted are most [[w:Ambiguity|ambiguous]]. 
=== Tea Leaves ===
[[File:Spring pouchong tea leaves on plate.jpg|thumb|''Spring Pouchong'' tea leaves that may be used for [[w:Tasseography|tasseography]] divination]]
[[w:Tasseography|Tasseography]] is the art of interpreting patterns in tea leaves, coffee grounds, or wine sediments. The diviner—a person skilled in interpreting tea leaves—looks at the pattern of tea leaves in the cup and allows the imagination to play around with the shapes suggested by them. They might look like a letter, a heart shape, a ring, or anything else. These shapes are then interpreted intuitively or by means of some system of symbolism. Images formed in a cup are created and uniquely seen by the reader, so it is often said that the only limitation for cup reading is the imagination of the reader themselves.

Tasseography reveals the essence of interpretation at an extreme. Because the tea leaf formations are entirely arbitrary the interpreter can say almost anything. The interpretation reflects judgements, evaluations, biases, concepts, models, analogies, predictions, optimism, pessimism, certainty, risk, doubt, hope, fears, storytelling skills, and desires of the interpreter, entirely independent of the tea leaf formation.

[[w:Rorschach_test|Rorschach tests]], [[w:Astrology|astrology]], [[w:Horoscope|horoscopes]], [[w:Biorhythm_(pseudoscience)|biorhythms]], [[w:Tarot_card_reading|tarot card reading]], [[w:Ouija|Ouija]], [[w:Fortune-telling|fortune telling]], and [[w:Dream_interpretation|dream interpretations]] are similar to tasseography in that they all depend more on the person doing the interpretation than on the ambiguous items being interpreted.

Recognizing the extent of ambiguity inherent in the various representations we use to describe reality, interpretation plays a significant role in influencing our understanding and beliefs. Adopting an interpretation can obscure and nearly occlude the underlying reality. Keep your eye on the territory as others offer various maps they would like you to use instead. Continue to reference reality so you can evaluate, challenge, and often reject, various interpretations.

=== Reality, Perception, and Interpretation ===
The parable of the [[w:Blind_men_and_an_elephant|blind men and the elephant]] provides an example that can help us examine the distinctions among [[w:Reality|''reality'']], [[w:Perception|''perception'']], and [[w:Interpretation_(philosophy)|''interpretation'']]. In the parable is a story of a group of blind men who have never come across an elephant before and who learn and imagine what the elephant is like by touching it. Each blind man feels a different part of the elephant's body, but only one part, such as the tail or the tusk. They then describe the elephant based on their limited experience and their descriptions of the elephant are different from each other.

The elephant is an example of ''reality''—what exists in the world. Each of the blind men makes some limited ''observations'' and forms a ''perception'' of the elephant. Each of these perceptions is based a small sample of reality. The man at the tail accurately perceives the tail by sensing how it feels. The man at the tusk accurately perceives the tusk, also by sensing how it feels. Then each man ''interprets'' his (limited) observation. The man at the tail concludes an elephant is like a rope, the man at the tusk concludes the elephant is hard, smooth, and like a spear.

Each blind man makes errors when interpreting their perceptions:
*Each fails to include the observation of the others, and if they consult the other observers, they fail to trust these additional observations and integrate them into a consistent whole. 
*Each [[w:Faulty_generalization|overgeneralizes]] from their limited experience to draw conclusions about the entire elephant.
*Each interprets their observations in terms of existing [[w:Paradigm|paradigms]] (rope, spear) rather than considering new paradigms (elephant). 

Recall the differing yet equally correct interpretations of the [[w:Rabbit–duck illusion|rabbit-duck illusion]].  This simple image allows for two different, yet equally correct interpretations. Life is often more complicated than ducks and rabbits, and we can reasonably differ in interpreting many words.

The [[w:Second_Amendment_to_the_United_States_Constitution|second amendment to the United States Constitution]] is the subject of long and contentious interpretations. It states:
&lt;blockquote&gt;
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
&lt;/blockquote&gt;
The (sometimes omitted) commas in the text draw much attention. The Militia clause is especially open to interpretation. Indeed, most of the words in this short statement have been interpreted in a variety of ways. Interpretation of this text is the subject of [[w:List_of_firearm_court_cases_in_the_United_States|many court cases]] and continues to sharply divide political discourse in the United States.

[[w:Innuendo|Innuendo]] and [[w:Plausible_deniability|plausible deniability]] [[w:Deception|disingenuously]] [[w:Equivocation|equivocate]] on interpretation.

Learn to [[Embracing_Ambiguity|embrace ambiguity]]. [[Finding_Common_Ground/Doubt_and_our_Bayesian_Brains|Become comfortable with doubt]]. Examine and reevaluate your preconceptions. Avoid tragic misinterpretations.

=== Tragic Interpretations ===
Interpretations that prematurely eliminate [[w:Doubt|doubt]] and resolve [[w:Ambiguity|ambiguity]] into a comfortable yet false feeling of [[w:Certainty|certainty]] have led to several tragedies. Here are some prominent examples.
#It has long been observed that the sky brightens each morning and day turn to darkness each evening. The causes of this were ambiguous for most of recorded history. Perhaps the sun moves around the earth on a [[w:Celestial_sphere|celestial sphere]]. Alternatively, the earth could rotate on its axis as it [[w:Heliocentrism|revolves around the sun]]. The pope was certain the earth was the center of the universe, [[w:Galileo_Galilei|Galileo]] differed. The contentious [[w:Galileo_affair|debate over these alternative interpretations]] of the observations culminated with the trial and condemnation of [[w:Galileo_Galilei|Galileo Galilei]] by the [[w:Roman_Inquisition|Roman Catholic Inquisition]] in 1633.
#In 1997 members of the [[w:Heaven's_Gate_(religious_group)#Mass_suicide|Heaven’s Gate]] new religious movement misinterpreted images of the Hale-Bopp comet, and decided that the only way to evacuate this earth was to participate in a mass suicide. [[w:Cult|Cults]] are often formed based on alternative interpretations of events.
#Although the phrase “[[w:All_men_are_created_equal|All men are created equal]]” motivated [[w:American_Revolutionary_War|the revolution]] that formed the United States, differing interpretations of the status of [[w:Slavery_in_the_United_States|slaves]] as being either property or being humans led to the [[w:American_Civil_War|civil war]]. 
#Religious groups often differ in the interpretation of various symbols, texts, and prophesies. Here are some examples.
##[[w:Christianity|Christianity]] is a religion based on interpretations of the [[w:Life_of_Jesus_in_the_New_Testament|life]] and [[w:Teachings_of_Jesus|teachings]] of [[w:Jesus|Jesus of Nazareth]]. These interpretations lead to the belief that [[w:Jesus|Jesus]] is the [[w:Son_of_God_(Christianity)|Son of God]], whose coming as the [[w:Messiah#Christianity|messiah]] was [[w:Old_Testament_messianic_prophecies_quoted_in_the_New_Testament|prophesied]] in the [[w:Hebrew_Bible|Hebrew Bible]] and chronicled in the [[w:New_Testament|New Testament]].
##The [[w:Judaism#Christianity_and_Judaism|differences between Christianity and Judaism]] originally centered on whether Jesus was the Jewish Messiah but eventually became irreconcilable. Followers of [[w:Judaism|Judaism]] interpret the life of Jesus to be that of a well-meaning and charismatic human who worked as carpenter, but not the Messiah. This difference in interpretation led to the [[w:The_Holocaust|holocaust]]. 
##[[w:Islam|Islam]] is a religion teaching that [[w:Muhammad|Muhammad is a messenger of God]]. The primary scriptures of Islam are the [[w:Quran|Quran]], interpreted as the verbatim word of God. [[w:Islam#Denominations|Various denominations]] differ in their interpretations of the rightful successors of Muhammad. This difference in interpretation has led to [[w:Human_rights_in_post-invasion_Iraq#Sectarian_warfare_in_Iraq|sectarian warfare]]. 
##[[w:Scientology|Scientology]], [[w:Scientology#Scientology_as_a_religion|classified as a religion]] by the United States Internal Revenue Service, is a set of beliefs and practices invented by American author [[w:L._Ron_Hubbard|L. Ron Hubbard]], and an associated movement. It has been variously defined as a [[w:Cult|cult]], a [[w:Scientology_as_a_business|business]], or a [[w:New_religious_movement|new religious movement]], depending on various interpretations of the various [[w:Scientology_beliefs_and_practices|beliefs and practices]].
##[[w:Nontheism|Nontheists]] study reality and recognize that [[Beyond_Theism#Non-Theism_is_the_Null_Hypothesis|non-theism is the parsimonious worldview]]. Therefore, theists who make supernatural claims bear the (unmet) burden of proving their supernatural claims. Nontheists may risk persecution for [[w:Blasphemy|blasphemy]]. 
##[[w:Religious_war|Religious wars]], often resulting from disputes over these various interpretations, are frequent, long lasting, and deadly. Matthew White's [[w:The_Great_Big_Book_of_Horrible_Things|''The Great Big Book of Horrible Things'']] gives religion as the primary cause of 11 of the world's 100 deadliest atrocities.
#The nature of the [[w:2021_United_States_Capitol_attack|2021 United States Capitol attack]] has been widely, passionately, and [[w:Domestic_reactions_to_the_2021_United_States_Capitol_attack|variously interpreted]]. Former attorney general [[w:William_Barr|William Barr]], who had resigned days earlier, denounced the violence, calling it &quot;outrageous and despicable&quot;, adding that the president's actions were a &quot;betrayal of his office and supporters&quot; and that &quot;orchestrating a mob to pressure Congress is inexcusable.&quot; None-the-less, the Republican National Committee contended that the lethal riot was an example of &quot;legitimate political discourse.&quot; [[w:2021_United_States_Capitol_attack#Aftermath|The aftermath]] continues to have important political, legal, and social repercussions.
#Various [[w:Conspiracy_theory|conspiracy theories]] are based on alternative interpretations of events. Here are a few selected from a much longer [[w:List_of_conspiracy_theories|list of conspiracy theories]].
##Many [[w:John_F._Kennedy_assassination_conspiracy_theories|conspiracy theories concerning the assassination of John F. Kennedy]] in 1963 have emerged. Many of these depend on various interpretations of the available evidence and are especially skeptical of the [[w:Single-bullet_theory|single bullet theory]]—the official description of the event appearing in the [[w:Warren_Commission|Warren commission report]]. It is also frequently asserted that the United States federal government intentionally covered up crucial information in the aftermath of the assassination to prevent the conspiracy from being discovered.
##The [[w:September_11_attacks|multiple attacks]] made on the US by [[w:Terrorism|terrorists]] using hijacked aircraft on September 11, 2001 have proven [[w:9/11 conspiracy theories|attractive to conspiracy theorists]]. Theories may include reference to missile or hologram technology. By far, the most common theory is that the attacks were in fact controlled demolitions, a theory which has been rejected by the engineering profession and the [[w:9/11_Commission|9/11 Commission]].
##The &quot;[[w:Deep_state_in_the_United_States|Deep state]]&quot; often refers to discredited allegations of an unidentified &quot;powerful elite&quot; who act in coordinated manipulation of a nation's politics and government. Proponents of such theories have included Canadian author [[w:Peter_Dale_Scott|Peter Dale Scott]], who has promoted the idea in the US since at least the 1990s, as well as [[w:Breitbart_News|''Breitbart News'']], [[w:Infowars|''Infowars'']] and former US President [[w:List_of_conspiracy_theories_promoted_by_Donald_Trump|Donald Trump]]. A 2017 poll by [[w:ABC_News|ABC News]] and The Washington Post indicated that 48% of Americans believe in the existence of a conspiratorial &quot;deep state&quot; in the US. Some of these theories promote [[w:QAnon|QAnon conspiracy theories]] which are based on the interpretation of false claims made by an anonymous individual or individuals known as &quot;Q&quot;.
##[[w:Anti-vaccination|Anti-vaccination activists]] and other people in many countries have spread a variety of unfounded [[w:COVID-19_vaccine_misinformation_and_hesitancy|conspiracy theories]] and other [[w:Misinformation|misinformation]] about [[w:COVID-19_vaccine|COVID-19 vaccines]] based on misinterpreted or misrepresented science, religion, exaggerated claims about side effects, a story about COVID-19 being spread by [[w:COVID-19_misinformation#5G_mobile-phone_networks|5G]], misrepresentations about how the immune system works and when and how COVID-19 vaccines are made, and other false or distorted information. This has prolonged the pandemic and caused political unrest. 
#[[w:Quackery|Quackery]], [[w:Quackery|crystal healing]], [[w:Homeopathy|homeopathy]], and other ineffective and fraudulent health claims waste time and money while deceiving patients and delaying effective treatments. These are sustained by inaccurate interpretation of evidence.
=== Assignment ===
#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of interpretations, language, perception, or reality.
##Identify the specific interpretations that differ.
##Identify the ambiguities that allow for various interpretations.
##Cast doubt on the certainty of any specific interpretation. 
#[[Practicing Dialogue|Practice dialogue]] rather than debate or argumentation. Be [[Candor|candid]]. [[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course.
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[w:Evaluating_Evidence|research them]].
##[[Embracing Ambiguity|Embrace ambiguity]].
##[[Seeking_True_Beliefs|Seek true beliefs]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Narration ==
[[File:John Everett Millais (1829-1896) - The Boyhood of Raleigh - N01691 - National Gallery.jpg|thumb|We are captivated by [[w:Storytelling|storytelling]].]]
Humans enjoy telling and retelling [[w:Narrative|stories]]. [[w:Myth|Myths]] have been a part of human culture for at least as long as recorded history. [[w:Folklore|Folklore]], [[w:Oral_tradition|oral traditions]], [[w:Epic_poetry|epics]], [[w:Creation_myth|creation myths]], [[w:Campfire_story|campfire stories]], [[w:Fairy_tale|fairy tales]], [[w:Legend|legends]], [[w:Bedtime_story|bedtime stories]], [[w:Song|songs]], [[w:Poetry|poems]], and [[w:Soap_opera|soap operas]] are told and retold. Stories provide a memorable, coherent, compelling, and plausible explanation for events. They are also often fanciful and factually unfounded.

In his book [[w:Sapiens:_A_Brief_History_of_Humankind|''Sapiens: A Brief History Of Humankind'']], author Yuval Harari claims that all large-scale human cooperation systems – including [[w:Religions|religions]], [[w:Political_structures|political structures]], [[w:Trade|trade networks]], and [[w:Legal_institutions|legal institutions]] – owe their emergence to Sapiens' distinctive cognitive capacity for [[w:Fiction|fiction]]. Ours is the storytelling species.

=== Narratives that stick ===
Why is it that some ideas survive while others die? According to the book [[w:Made_to_Stick|''Made to Stick'']], an idea becomes memorable or interesting when it is: 
*Simple – find the core of any idea or thoughts
*Unexpected – grab people's attention by surprising them
*Concrete – make sure an idea can be grasped and remembered later
*Credible – give an idea believability and credibility
*Emotional – help people see the importance of an idea
*Presented as a story – empower people to use an idea through narrative

Several grand narratives currently divide our [[Exploring Worldviews|worldviews]] and political discourse in the United States. [[w:Conservatism|Conservatives]] often agree with Ronald Regan that “[[w:Ronald_Reagan#First_Inaugural_address_(1981)|Government is the Problem]]”, while [[w:Progressivism|progressives]] tend to believe that “Government is the solution”. Each [[w:Tribe|tribe]] can cite many examples bolstering their position. Beginning with their chosen narrative, conservatives often [[w:Mask_refusal|oppose mask mandates]], while progressives blame the unmasked for endangering others and prolonging the pandemic. These arguments begin with narratives and extend to interpretations and representative symbols but are rarely based on a careful [[Evaluating_Evidence|evaluation of evidence]].

Americans are divided by several such narratives. More [[w:Gun_politics_in_the_United_States|guns make us safer]], or they cause needless [[w:Gun_violence_in_the_United_States|violence]]. [[w:Climate_change|Climate change]] is the biggest threat to our future or is simply a hoax. [[w:Abortion_in_the_United_States|Abortion]] murders babies or protects a women’s constitutional right to choose. God created man, or [[w:Problem_of_the_creator_of_God|man created God]]. [[w:Psalm_115|Earth belongs to man]] or [[q:Chief_Seattle|man belongs to earth]]. Capitalism is the solution or [[w:Criticism_of_capitalism|capitalism is the problem]]. Do you believe the experts or do you [[w:Knowing_How_You_Know/Divided_by_epistemology|believe your friends]]? Various conspiracy theories provide especially troublesome narratives. Ideologies amplify [[w:List_of_cognitive_biases|cognitive biases]].
=== Powerful False Narratives ===
Very often, [[w:Storytelling|the best story wins]]. Here is an example of a powerful, influential, and harmful false narrative, known as the ''satanic panic''.

The [[w:Satanic_panic|Satanic panic]] is a [[w:Moral_panic|moral panic]] consisting of over 12,000 unsubstantiated cases of Satanic ritual abuse (SRA) starting in the United States in the 1980s, spreading throughout many parts of the world by the late 1990s, and persisting today. The panic originated in 1980 with the publication of [[w:Michelle_Remembers|''Michelle Remembers'']], a bestselling book co-written by Canadian psychiatrist [[w:Lawrence_Pazder|Lawrence Pazder]] and his patient (and future wife), Michelle Smith, which used the discredited practice of [[w:Recovered-memory_therapy|recovered-memory therapy]] to make sweeping lurid claims about satanic ritual abuse involving Smith. The allegations which afterwards arose throughout much of the United States involved reports of [[w:Physical_abuse|physical]] and [[w:Sexual_abuse|sexual abuse]] of people in the context of [[w:Occult|occult]] or [[w:Theistic_Satanism|Satanic]] rituals. In its most extreme form, allegations involve a conspiracy of a global Satanic cult that includes the wealthy and powerful world elite in which children are abducted or bred for [[w:Human_sacrifice|human sacrifices]], [[w:Child_pornography|pornography]], and [[w:Prostitution|prostitution]].

The key elements of the narrative are:
*Satanic ritual abuse is horrific and widespread.
*Unknown to us many of our children are being subjected to the horrors of satanic ritual abuse.
*The abuse is so horrible that the children [[w:Repressed_memory|repress their memories]] of the abuse and are unable to disclose their experiences.
*A newly developed interviewing technique, called [[w:recovered-memory_therapy|recovered-memory therapy]], can elicit accurate memories and testimony from the children.
*Using this technique, many children are beginning to reveal and describe the abuse they have suffered.
*This must be urgently investigated, and the abuse stopped at all costs.
*A global Satanic cult may be responsible.
*Missing memories among the victims and absence of evidence was cited as evidence of the power and effectiveness of the cult in furthering their agenda.

Initial interest arose via the publicity campaign for Pazder's 1980 book [[w:Michelle_Remembers|''Michelle Remembers'']], and it was sustained and popularized throughout the decade by coverage of the [[w:McMartin_preschool_trial|McMartin preschool trial]] and the contemporaneous [[w:Day-care_sex-abuse_hysteria|day-care sex-abuse hysteria]]. Testimonials, symptom lists, rumors, and techniques to investigate or uncover memories of SRA were disseminated through professional, popular, and religious conferences, as well as through [[w:Talk_show|talk shows]], sustaining and further spreading the moral panic throughout the United States and beyond. In some cases, allegations resulted in criminal trials with varying results; after seven years in court, the McMartin trial resulted in no convictions for any of the accused, while other cases resulted in lengthy sentences, some of which were later reversed. Scholarly interest in the topic slowly built, eventually resulting in the conclusion that the phenomenon was a moral panic, which, as one researcher put it in 2017, &quot;involved hundreds of accusations that devil-worshipping pedophiles were operating America's white middle-class suburban daycare centers.&quot; 

Of the more than 12,000 documented accusations nationwide, investigating police were not able to substantiate any allegations of organized cult abuse.

Today the [[w:Radical_right_(United_States)|far-right]] conspiracy theory movement known as [[w:QAnon|QAnon]], has adopted many of the tropes of Satanic Ritual Abuse and Satanic Panic. Instead of daycare centers being the center of abuse, however, liberal [[w:Hollywood|Hollywood]] actors, [[w:Democratic_Party_(United_States)|Democratic]] politicians, and high-ranking government officials are portrayed as a child-abusing cabal of Satanists.
=== Assignment ===
'''Part 1:'''

#Identify a powerful false narrative to study for this assignment. Choose one from this list of [[Finding Common Ground/Powerful False Narratives|powerful false narratives]], or from any other source.
#Identify the elements that make this narrative compelling, convincing, memorable, and likely to spread and be shared with others.
#Identify the falsehoods in the narrative.
#Identify any [[Embracing Ambiguity|ambiguities]] that are prematurely resolved. 
#Identify the various calls to action inspired by the narrative.
#How is this narrative harmful, if at all? 
#Who gains and who loses as this narrative spreads?

'''Part 2:'''

#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of narrative, language, perception, or reality.
#If the narrative is driving the conversation, go through the steps above to analyze the narrative elements.
#Practice dialogue rather than debate or argumentation. Be [[w:Candor|candid]]. [[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course. 
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[Evaluating Evidence|research them]].
##[[Seeking True Beliefs|Seek true beliefs]].
##[[Embracing Ambiguity|Embrace ambiguity]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Ideology ==
[[File:Ideology Occludes Reality.jpg|thumb|Ideology Occludes Reality]] 
Do you think for yourself and choose your own beliefs? If you are like most people, you probably find it easier to adopt some pre-packaged set of beliefs that seem attractive. We can broadly characterize an [[w:Ideology|ideology]] as some agenda-driven set of beliefs.&lt;ref&gt;Many definitions of ideology are proposed. See, for example: [https://journals.sagepub.com/doi/10.1177/106591299705000412 Ideology: A definitional Analysis], John Gerring, December 1, 1997, Political Research Quarterly. &lt;/ref&gt;  Because ideologies are constructed to serve some agenda, an ideology is unlikely to accurately represent reality. 

An [[w:ideology|ideology]] is a set of beliefs intended to describe how the world works, or how some believe it should work. An ideology is a particular way of looking at the world, often codified into a [[w:doctrine|doctrine]]. Often our religious, political, and economic beliefs are drawn from an ideology. You may also follow particular lifestyle choices such as [[w:veganism|veganism]], or [[w:Environmentally_friendly|environmentalism]] based on a particular ideology. Ideologies substitute socially constructed models for brute facts. Many ideological models do not correspond well to reality. “Essentially, all models are wrong,” [[w:George_E._P._Box|George Box]] noted, “but some are useful.” Beware of substituting an ideology for a careful examination of reality.

As illustrated in the revised diagram shown here, ideologies impose a model that prohibits direct access to reality, and displaces any alternative narratives, interpretations, representations, or perceptions of reality. The ideology establishes all you need to know. It acts as a convenient substitute for reality. 

Immersion and commitment to an ideology can become a firmly held part of your identity. If you say “I am a Conservative” rather than “I often agree conservative political ideas” you are declaring the ideology as a part of your identity.&lt;ref&gt;{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant|date=February 2, 2021 |title=Think Again: The Power of Knowing What You Don't Know  |publisher=Viking |pages=320 |isbn=978-1984878106}}&lt;/ref&gt; This can make it harder to abandon. Although you choose your beliefs, [[True_Self#I_am|you are not your beliefs]]. 

Remaining bound by an ideological doctrine is a form of mental bondage. It is wise to break free from that bondage. Adopting a scout mindset—described below—can help us break free from ideologies that are holding our minds captive.

=== The Scout Mindset ===
[[File:U.S. Marine and Japan Ground Self-Defense Scout Snipers 170310-M-PQ336-010.jpg|thumb|Scouts seek to see the world as it is, not as they wish it was.]]
Author [[w:Julia_Galef|Julia Galef]] describes the ''scout mindset'' as “The motivation to see things as they are, not as you wish they were.”&lt;ref&gt; {{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}} @9 of 541&lt;/ref&gt;  In contrast to the ''scout mindset'', the ''soldier mindset'' is a motivation to attack differing points of view or defend a position in each argument we encounter. 
 
Contrasted with the soldier mindset, the scout mindset values:
*'''Seeing things as they''' are over defending territory or taking territory;
*'''Asking is it true?''' over arguing to defend a pre-determined position;
*'''Observations''' over interpretations;
*'''Investigation''' over argumentation;
*'''Gaining insight''' over winning this argument;
*'''Wonder''' over attacking;
*'''Learning''' over advancing a position;
*'''Observing''' over taking ground;
*'''Understanding the issue''' over shooting down arguments;
*'''Exploring possibilities''' over reinforcing a position;
*'''[[Seeking True Beliefs|Seeking true beliefs]]''' over securing long held beliefs;
*'''Seeking reality''' over defending an ideology;
*'''[[Evaluating Evidence|Objective evidence]]''' over motivated reasoning;
*'''Representative evidence''' over narratives, specious interpretations and representations;
*'''Reason''' over rhetoric;
*'''Exploration''' over staying on the ideological course;
*'''Dialogue''' over reciting [[w:Dogma|dogma]];
*'''Listening''' over reiterating and continuing to advocate;
*'''[[Virtues/Humility|Humility]]''' over arrogance;
*'''Curiosity''' over certainty or fear;
*'''[[Intellectual honesty]]''' over [[w:Prevarication|prevarication]], and
*'''Working with collaborators''' over fighting with opponents.

[[Coping with Ego|Ego]] encourages the soldier. [[Deductive_Logic/Clear_Thinking_curriculum|Reason]] encourages the scout.

The [[w:Stanford_Encyclopedia_of_Philosophy|Stanford Encyclopedia of Philosophy]] entry on Law and Ideology tells us, “Ideologies are ideas whose purpose is not epistemic, but political. Thus, an ideology exists to confirm a certain political viewpoint, serve the interests of certain people, or to perform a functional role in relation to social, economic, political, and legal institutions.” &lt;ref&gt;Sypnowich, Christine, &quot;[https://plato.stanford.edu/archives/sum2019/entries/law-ideology Law and Ideology]&quot;, The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.) &lt;/ref&gt;

Because ideologies exist to advance an agenda rather than to model reality, they require a soldier mindset to sustain the gap between the model described by the ideology and an accurate model based on reality. Ideologies are vulnerable to exploration by a scout mindset. Adopting a scout mindset is key to escaping ideologies.

=== Example Ideologies ===
We rely on many ideologies to simplify our thinking. These include: 
*[[w:Loyalty|Loyalty]], including loyalty to an individual person, a group, team, brand, causes, regions, nation, or any of the belief systems listed below. 
*[[w:List_of_mythologies|Myths and legends]],
*[[w:Folklore|Folklore]], [[w:Tradition|traditions]], [[w:Superstition|superstitions]], and [[w:Taboo|taboos]],
*[[w:Caste|Caste systems]], including [[w:Racism|racism]], [[w:Sexism|sexism]], other forms of [[w:Ascribed_status|ascribed status]], [[w:Social_class|social classes]], and other ranking constructs, 
*[[w:Stereotype|Stereotypes]], 
*[[w:Economic_ideology|Economic ideologies]], including [[w:Neoliberalism|neoliberalism]], [[w:Monetarism|monetarism]], [[w:Mercantilism|mercantilism]], [[w:Mixed_economy|mixed economy]], [[w:Social_Darwinism|social Darwinism]], [[w:Communism|communism]], [[w:Laissez-faire|laissez-faire economics]], and [[w:Free_trade|free trade]]. There are also current theories of [[w:Safe_trade|safe trade]] and [[w:Fair_trade|fair trade]] that can be understood as ideologies.
*[[w:List_of_political_ideologies|Political ideologies]], including anarchism, authoritarianism, communism, conservatism, democracy, environmentalism, fascism, separatist movements, liberalism, libertarianism, nationalism, populism, social democracy, socialism, and others.
*[[w:List_of_creation_myths|Creation myths]],
*[[w:List_of_religions_and_spiritual_traditions|Religions and spiritual traditions]], 
*[[w:Quackery|Quackery]],
*[[w:List_of_topics_characterized_as_pseudoscience|Pseudoscientific beliefs]],
*[[w:Paranormal|Paranormal beliefs]],
*Commitment to [[w:List_of_conspiracy_theories|conspiracy theories]], and
*[[w:List_of_new_religious_movements|New religious movements]].
=== Assignment ===
'''Part 1:'''

#Identify each of the ideologies you identify with, belong to, or agree with. Use the list above as a guide or use any other method to identify ideologies that influence you.
#For each of the ideologies identified in step 1:
##Decide if the model it presents is an accurate representation of reality in all its scope and complexity.
##Identify any [[Embracing Ambiguity|ambiguities]] that are prematurely resolved. 
##Decide if the ideology is helping you understand reality, or is occluding, limiting, biasing, or censoring your understanding of reality.
##If you decide a particular ideology is unhelpful, take steps to abandon that ideology. 

Welcome those who are [[Seeking True Beliefs|seeking truth]]. Abandon those who are [[w:Certainty|certain]] they have found [[w:Dogma|Truth]].

'''Part 2:'''
#Study the module on [[Knowing_How_You_Know/Examining_Ideologies|Examining Ideologies]] within the [[Knowing How You Know|Knowing how you know]] course.
#Complete the [[Knowing_How_You_Know/Examining_Ideologies#Assignment|assignments]] in that module.
#Read the essay [[/Every Ism Creates a Schism/|&quot;Every Ism Creates a Schism&quot;: An Exploration]].
#Think beyond the doctrine.

== Toward Ought ==
[[File:Compass rose browns 00.png|thumb|right| 250px|A deep understanding of [[Moral_Reasoning#A_Basis_for_Moral_Reasoning |''impartiality'']] can guide toward what we ought to do. ]]
So far in this course, the common ground we have considered is our shared reality. This is our collective understanding of ''what is'' in the world. Can we find a corresponding common ground regarding what we ''ought to do''?

Philosopher [[w:David_Hume|David Hume]] famously observed that knowing only ''what is'', we cannot determine what we [[w:Is–ought problem|''ought to do'']]. However, by making the reasonable [[Moral_Reasoning#A_Basis_for_Moral_Reasoning |assumption of ''impartiality'']], we can begin to identify what it is we ought to do. 

We ought to [[Assessing_Human_Rights/Beyond_Olympic_Gold|advance human rights worldwide]], adopt [[Level_5_Research_Center#Values|pro-social values]], and establish a well-founded basis for [[Moral Reasoning|moral reasoning]].

=== Assignment ===
#Complete the Wikiversity course on [[Assessing Human Rights]].
#[[Assessing_Human_Rights/Beyond_Olympic_Gold|Advance human rights, worldwide]]. 
#Study this [[Level_5_Research_Center#Values|list of pro-social values]].
#Adopt pro-social values and [[Level_5_Research_Center/Choosing_Level_5_Living|choose level 5 living]]. 
#Complete the Wikiversity course on [[Moral Reasoning]].
#Develop your own well-founded basis for [[Moral Reasoning|moral reasoning]].

== Simple but not easy ==
The central idea in this course—[[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]]—is simply stated and supported by overwhelming evidence. However, it may not be easy for you, or others you encounter or care about to change deeply held beliefs that differ from objective reality. Here are some steps you can take to align your beliefs with reality and work to find common ground with others.
#Begin by aligning your own beliefs with reality. Complete the course on [[Seeking True Beliefs]]. [[Exploring_Worldviews/Aligning_worldviews|Align your own worldview]] with reality. [[Deductive_Logic/Clear_Thinking_curriculum|Think clearly]]. [[Knowing How You Know|Know how you know]]. Become skillful at [[Evaluating Evidence|evaluating evidence]]. Stay curious. [[Embracing Ambiguity|Embrace ambiguity]].
#Be careful to [[Facing_Facts#Degrees_of_Consensus|distinguish among matters of fact]], matters of controversy, and matters of opinion. Do not argue matters of fact, research them. Do not argue matters opinion, enjoy them. Reason carefully and listen closely when discussing matters of controversy. Although [[Virtues/Tolerance|tolerance]] is essential in matters of opinion, it has no place in [[Facing Facts|matters of fact]].
#Spend the required effort to prepare to find common ground with others. 
##Complete the course on [[Practicing Dialogue|practicing dialogue]]. Practice dialogue.
##Ensure all participants have adopted a [[Socratic_Methods#Essential_Socratic_Temperament|Socratic temperament]].
##Expect [[intellectual honesty]].
###Complete the course on [[intellectual honesty]].
###[[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
###Discuss the importance of [[intellectual honesty]]—accurately describing your true beliefs—with each participant. Gain agreement to be intellectually honest, and to expect intellectual honesty from all participants. Do not indulge [[w:Charlatan|charlatans]]. Do not [[w:Internet_troll|feed the trolls]]. 
#Finding common ground often challenges deeply held beliefs. This is uncomfortable; people often react with fear and may seek to withdraw from the dialogue or divert the conversation.  Notice the shift from [[Finding_Common_Ground#The_Scout_Mindset|scout mindset]] to soldier mindset. Notice when fear begins to displace curiosity. Encourage the fearful person to allow curiosity to displace fear. Return to a scout mindset.
#Real good is our common ground. People who are [[Living Wisely/Seeking Real Good|seeking real good]] will find common ground with others who are also seeking real good. When [[Transcending Conflict|conflict arises]], it is likely because someone is not seeking real good. 
#The course on [[Street Epistemology]] provides specific techniques for exploring the basis for beliefs. Use applicable Street Epistemology techniques when seeking common ground.

== Summary and Conclusions ==
Reality exists. Reality is vast, complex, and dynamic. Humans have only investigated a small portion of the universe, and our investigation is incomplete. We all live together on this one planet we call Earth and [[Knowing_How_You_Know/One_World|we all live in the same universe]].

Because we all live in the same universe, our reliable understanding of that universe must eventually converge toward one coherent description. Because reality exists, we can [[Evaluating_Evidence|examine reality]], and we can [[Exploring_Worldviews/Aligning_worldviews|align our worldview]] with reality. Building on these observations, it follows that [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]], even though any number of worldviews is possible.

[[w:Perception|Perception]] transforms sensory information originating with some material object, known as the ''target'' or the ''[[w:Distal|distal]] stimulus'' into mental representations known as [[w:Perception#Process_and_terminology|percepts]], which do not exist elsewhere in the world. 

[[w:Perception|Perception]]—the process of extracting information from energy that impinges on our [[w:Sense|sensory organs]]—is not straightforward. There is more to perception than meets the eye.

Our most direct encounters with reality are through direct sensory observations and awareness. However, our perceptions are not exact replicas of the distal objects being perceived. Omissions, distortions, and additions take place. Fortunately, we can augment our direct perceptions to obtain a more accurate understanding of our world. Furthermore, [[Facing_Facts/Perceptions_are_Personal|perceptions are personal]], and it is often a [[w:Problem_of_induction|mistake to generalize]] our personal perceptions beyond our own experiences.

We represent our thoughts using various symbols, including words, gestures, facial expressions, images, and analogies. The mapping from various representations to reality is imprecise. Language and other symbols can be constraining, ambiguous, refer to social constructs, prejudiced, based on artificial boundaries, or focus on convenient representation rather than reality. The map is not the territory, although we often confuse our representations for realty.

People [[w:Interpretation_(philosophy)|interpret]] observations to resolve ambiguity, increase certainty, and to account for observations within some familiar model or analogy. The role of interpretation is most evident when the items being interpreted are most [[w:Ambiguity|ambiguous]]. 

[[w:Tasseography|Tasseography]]—reading tea leaves—reveals the essence of interpretation. Because the tea leaf formations are entirely arbitrary the interpreter can say almost anything. The interpretation reflects judgements, evaluations, biases, concepts, models, analogies, predictions, optimism, pessimism, certainty, risk, doubt, hope, fears, storytelling skills, and desires of the interpreter, entirely independent of the tea leaf formation.

The [[w:Rabbit–duck illusion|rabbit-duck illusion]] allows for two different, yet equally correct interpretations. Life is often more complicated than ducks and rabbits, and we can reasonably differ in interpreting many words.

Interpretations that prematurely eliminate [[w:Doubt|doubt]] and resolve [[w:Ambiguity|ambiguity]] into a comfortable feeling of [[w:Certainty|certainty]] have led to several tragedies. Become [[Embracing Ambiguity|comfortable with ambiguity]].

Humans enjoy telling and retelling [[w:Narrative|stories]]. This may be the defining characteristic of the human species. We are exposed to many powerful false narratives. To find common ground we must dismiss the falsehoods in narratives.

An [[w:ideology|ideology]] is a set of beliefs intended to describe how the world works, or how some believe it should work. An ideology is a particular way of looking at the world, often codified into a [[w:doctrine|doctrine]]. Often our religious, political, and economic beliefs are drawn from an ideology.

Ideologies substitute socially constructed models for brute facts. Many ideological models do not correspond well to reality. “Essentially, all models are wrong,” [[w:George_E._P._Box|George Box]] noted, “but some are useful.” Beware of substituting an ideology for a careful examination of reality.

Remaining bound by an ideological doctrine is a form of mental ''bondage''. It is wise to break free from that bondage. Adopting a ''scout mindset'' can help us break free from ideologies that are holding our minds captive.

It is likely your beliefs are influenced by [[Knowing How You Know/Examining Ideologies|inaccurate ideologies]]. Identify these and abandon them.

This is not about compromise. This is about gaining an accurate understanding of the world we live in.

[[Facing Facts/Reality is the Ultimate Reference Standard|Reality is our reference standard]]. 

Embrace reality. Seek true beliefs. Dismiss misleading perceptions, representations, interpretations, narrations, and ideologies. Align concepts with reality.

[[Assessing_Human_Rights/Beyond_Olympic_Gold|Advance human rights worldwide]], adopt [[Level_5_Research_Center#Values|pro-social values]], and establish a well-founded basis for [[Moral Reasoning|moral reasoning]].

[[Transcending_Conflict|Transcend conflict]].

[[Living_Wisely/Seeking_Real_Good|Seek real good]].

It is difficult to change deeply held beliefs, however if ''you'' can do this, ''they'' can do this.

We can [[Facing_Facts/Reality_is_our_common_ground|find common ground]].

== Recommended Reading ==
*{{Cite book|title=Truth: what it is, how to find it, and why it still matters|publisher=Johns Hopkins University Press|date=2026|location=Baltimore|isbn=978-1-4214-5372-9|first=Michael|last=Shermer}}
*{{cite book |last=Van der Stigchel |first=Stefan |date=March 12, 2019 |title=How Attention Works: Finding Your Way in a World Full of Distraction |publisher=The MIT Press |pages=152 |isbn=978-0262039260 }}
*{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }} 
*{{cite book |last1=Gause  |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is  |publisher=Dorset House Publishing Company |pages=176 |isbn=978-0932633163 |author-link=w:Gerald_Weinberg}} 
*{{cite book |last=Weinberg |first=Gerald M. |author-link=w:Gerald_Weinberg |date=September 1, 1989 |title=Exploring Requirements: Quality Before Design  |publisher=Dorset House Publishing Company |pages= 320 |isbn=978-0932633132}}
*{{cite book |last=Holmes |first=Jamie |author-link=w:Jamie_Holmes_(author) |date=October 11, 2016 |title=Nonsense: The Power of Not Knowing Paperback  |publisher=Crown |pages=336 |isbn=978-0385348393}}
*{{cite book |last=Ariely |first=Dan |author-link=w:Dan_Ariely |date=September 17, 2024 |title=Misbelief: What Makes Rational People Believe Irrational Things |publisher=Harper Perennial  |pages=320 |isbn=978-0063280434}}
*{{cite book |last=Burton M.D. |first=Robert A.  |date=Mar 17, 2009 |title=On Being Certain: Believing You Are Right Even When You're Not |publisher=St. Martin's Griffin |pages272 |isbn=978-0312541521}}
*{{cite book |last=Duke |first=Annie |author-link=w:Annie_Duke |date= |title=Thinking in Bets: Making Smarter Decisions When You Don't Have All the Facts |publisher=Portfolio |pages= 288 |isbn=978-0735216372}}
*{{cite book |last=Freinacht  |first=Hanzi  |date=March 10, 2017 |title=The Listening Society: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=414 |isbn=978-8799973903}}
*{{cite book |last=Freinacht  |first=Hanzi  |date=May 29, 2019 |title=Nordic Ideology: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=495 |isbn=978-8799973927}}
*{{cite book |last1=Gilovich |first1=Thomas |last2=Ross |first2=Lee |date=December 1, 2015 |title=The Wisest One in the Room: How You Can Benefit from Social Psychology's Most Powerful Insights|publisher=Free Press|pages=320|isbn=978-1451677546}} 
*{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}}
*{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}
*{{cite book |last1=Hofstadter |first1=Douglas R  |last2=Sander |first2=Emmanuel |author-link=w:Douglas_Hofstadter |date=April 23, 2013 |title=Surfaces and Essences: Analogy as the Fuel and Fire of Thinking  |publisher=Basic Books |pages=592 |isbn=978-0465018475}}
*{{cite book |last= Weinberg |first=Gabriel  |date=June 18, 2019 |title=Super Thinking: The Big Book of Mental Models |publisher=Portfolio |pages=352  |isbn=978-0525533580}}
*{{cite book |title=The Art of Possibility: Transforming Professional and Personal Life |last1=Stone Zander |first1=Rosamund |last2=Zander|first2=Benjamin   |year=224 |publisher=Penguin |isbn=978-0142001103  |pages=224}}
*{{cite book |last1=Lakoff |first1=George |last2=Johnson |first2=Mark|date=April 15, 2003 |title=Metaphors We Live By |publisher= |pages=242 |isbn=978-0226468013 |author-link=w:George Lakoff }}
*{{cite book |last1=Heath |first1=Chip |last2=Heath |first2=Dan  |author-link=w:Chip_Heath|date= |title=Made to Stick: Why Some Ideas Survive and Others Die   |publisher=Random House|pages= 291 |isbn=978-1400064281}} 
*{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant|date=February 2, 2021 |title=Think Again: The Power of Knowing What You Don't Know  |publisher=Viking |pages=320 |isbn=978-1984878106}} 
*{{cite book |last1=Campbell |first1=Joseph | last2=Moyers |first2=Bill |date=June 1, 1991 |title=The Power of Myth  |publisher=Anchor |pages=293 |isbn=978-0385418867 |author-link=w:Joseph_Campbell }}
*{{cite book |last=Mackay |first=Charles |date=November 1, 2016 |title=[[w:Extraordinary Popular Delusions and The Madness of Crowds|Extraordinary Popular Delusions and The Madness of Crowds]] |publisher=CreateSpace Independent Publishing Platform |pages=386 |isbn=978-1539849582 |author-link=w:Charles_Mackay_(author) }}
*{{cite book |last=Orwell |first=George  |date=March 1, 2017 |title=[[w:Animal Farm|Animal Farm]] |publisher=Fingerprint! Publishing |pages=152 |isbn=978-9386538284 |author-link=w:George_Orwell }}
*{{cite book |last=Orwell  |first=George |date=January 15, 2013 |title=[[w:Politics and the English Language|Politics and the English Language]] |publisher=Penguin Classic |pages=48 |isbn=978-0141393063 |author-link=w:George_Orwell }}
*{{cite book |last=Hecht |first=Jennifer Michael |author-link=w:Jennifer_Michael_Hecht |date=September 7, 2004 |title=Doubt: A History: The Great Doubters and Their Legacy of Innovation from Socrates and Jesus to Thomas Jefferson and Emily Dickinson |publisher=HarperOne |pages=576 |isbn=978-0060097950}}
*{{cite book |last=Carroll |first=Sean M |author-link=w:Sean_M._Carroll |date=May 4, 2017 |title=The Big Picture: On the Origins of Life, Meaning, and the Universe Itself  |publisher=Oneworld Publications |pages=480 |isbn=978-1786071033}}
*{{cite book |last=Pinker |first=Steven |author-link=w:Steven_Pinker |date=February 13, 2018 |title=Enlightenment Now: The Case for Reason, Science, Humanism, and Progress |publisher=Viking |pages=576  |isbn=978-0525427575}}
*{{cite book |last=Wilczek |first=Frank |author-link=w:Frank_Wilczek |date=January 12, 2021 |title=Fundamentals: Ten Keys to Reality  |publisher=Penguin Press |pages=272 |isbn=978-0735223790}}
*{{cite book |last=Gray |first=Dave |author-link= |date=September 14, 2016 |title=Liminal Thinking: Create the Change You Want by Changing the Way You Think  |publisher=Two Waves Books |pages=184 |isbn=978-1933820460}}
*{{cite book |last=Schulz |first=Kathryn |author-link=w:Kathryn_Schulz |date=June 8, 2010 |title=Being Wrong: Adventures in the Margin of Error |publisher=Ecco |pages=416 |isbn=0061176044}}
*{{cite book |last=Temple |first=David J.  |date=April 2, 2024 |title=First Principles and First Values: Forty-Two Propositions on Cosmoerotic Humanism, the Meta-Crisis, and the World to Come  |publisher=World Philosophy &amp; Religion Press |pages=296 |isbn=979-8989588909}}

I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research.

*{{cite book |last=Coleman |first=Peter T. |author-link=w:Peter_T._Coleman_(academic) |date=June 1, 2021 |title=The Way Out: How to Overcome Toxic Polarization  |publisher=Columbia University Press |pages=296 |isbn=978-0231197403}}

== References ==
&lt;references/&gt;

{{CourseCat}} 
[[Category:Life skills]]
[[Category:Applied Wisdom]]
[[Category:Philosophy]]
[[Category:Clear Thinking]]
[[Category:Courses]]
[[Category:Community]]
[[Category:Reality]]
[[Category:Reformation Workshop]]
{{Clear Thinking}}</text>
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      <text bytes="101341" sha1="gxnk1gxjvamtexb8p31ouqaxkmgbpvv" xml:space="preserve">— Aligning concepts with reality.
[[File:Layers of Abstraction.jpg|thumb|We engage reality at various layers of abstraction.&lt;ref&gt; This diagram is used as the primary organizing structure for the course. Each ring in the diagram corresponds to a section of the course. Notice that each boundary is blurred. This blurring acknowledges that the various layers interact and no sharp boundary between layers exists. &lt;/ref&gt;]] 
{{TOC right | limit|limit=2}} 
== Introduction ==
Although we all live in the same world and share a single reality, we often seem to be worlds apart when discussing important issues. What is going on? How can we find common ground?&lt;ref&gt;According to [[w:Aumann's_agreement_theorem|Aumann's agreement theorem]], we will be able to find common ground.&lt;/ref&gt;
== Objectives ==
{{100%done}}{{By|lbeaumont}}
The objectives of this course are to:
*Understand the nature of reality.
*Identify the many layers of abstraction through which we encounter reality.
*Navigate through these layers of abstraction.
*Diagnose reasons for conflict during discussions.
*Find common ground.

This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]], the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking curriculum]], and the [[Coming Together]] curriculum.

Use this [[/Daily Practice: Finding Common Ground/|daily practice checklist]] to make finding common ground a habit. 

[https://www.academia.edu/106815969/Finding_Common_Ground Slides based on this course] are available, along with a [https://youtu.be/pQMEDRVwI5U video presentation] of those slides. 

If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont|click here to send me an email]] or leave a comment or question on the [[Talk: Finding Common Ground|discussion page]].

[[File:Finding Common Ground Audio Dialogue.wav|thumb|Finding Common Ground Audio Dialogue]]

== Importance ==
Finding common ground is an important skill because it is useful for resolving the [[w:Social_polarization|social]], [[w:Culture_war|cultural]], [[w:Political_polarization|political]], and [[w:Economic_inequality|economic]] polarization that is prevalent. [[w:Religious_violence|Religious conflicts]], [[w:War|international conflicts]], [[w:Nationalism|nationalism]], [[w:Class_conflict|class struggles]], [[w:Ethnic_conflict|racial tensions]], [[w:Culture_war|culture wars]], ideological conflicts, [[w:Gender_inequality|gender inequality]], [[w:Political_polarization|political polarization]], [[w:Book_censorship|book banning]], [[w:Hate_crime|hate crimes]], [[w:Climate_change_denial|climate change denial]], and [[w:Conspiracy_theory|conspiracy theories]] are a few examples of the conflicts that are dividing people throughout the world.

When we can recognize that [[Facing Facts/Reality is our common ground|reality is our common ground]], we can restore the [[Virtues/Civility|civility]] that is essential to peaceful coexistence. 

== In a Nutshell ==
Here is a brief introduction to the key concepts presented in this course. [[Finding_Common_Ground#Introduction|The diagram]] can help you organize, recall, share, and apply these ideas.
#[[Finding_Common_Ground#Reality|Reality]] exists and we all share in a single objective reality.&lt;ref&gt;{{Cite web|url=https://substack.com/inbox/post/174781036|title=A Minimum Viable Metaphysics|last=Substack|website=substack.com|language=en|access-date=2025-09-30}}&lt;/ref&gt;
#There are several reasons why this fact is difficult for us to grasp and hold onto.
##[[Finding_Common_Ground#Reality|Reality]] is vast, complex, and dynamic. Each of us directly encounters only a tiny slice of reality. We each experience only a glimpse of our vast universe.
##Our direct contact with reality is through our [[Finding_Common_Ground#Perception|perceptions]], which introduce omissions, distortions, and additions.
##Although we use words and other symbols to [[Finding_Common_Ground#Representation|represent]] reality as we perceive it, these symbols are limited, and they only provide approximate representations of our perceptions.
##Because much of what we encounter is [[Embracing Ambiguity|ambiguous]], it invites us to [[Finding_Common_Ground#Interpretation|interpret]] the information to resolve the ambiguity and provide us the comfort of certainty. Many cognitive biases influence our interpretations.
##We love telling and retelling [[Finding_Common_Ground#Narration|stories]]. We easily substitute alluring stories for the complexities and difficulties of reality.
##[[Finding_Common_Ground#Ideology|Ideologies]] substitute a simplified belief system for the complexities of reality. We are easily attracted to these easy to use explanations.
#We can see beyond these illusions and better comprehend reality.
#Reality is our common ground. We can each find that common ground by advancing toward the center of [[Finding_Common_Ground#Introduction|the diagram]] shown above.
#Although these ideas are simply stated, they are difficult to fully grasp and put into practice. Please complete the remainder of this course and use these insights every day.
#We can find common ground.

== Reality ==
For many practical reasons, this course begins with the assumption that [[w:Reality|reality]] exists. Everyday experience provides [[Evaluating Evidence|evidence]] that reality exists. Every time you decide to open the door before passing through the doorway, you are betting that reality exists. If you have lost your keys, then opening the door, or starting your car can become a real problem. If you have difficulty levitating, leaping tall buildings in a single bound, seeing through brick walls, teleporting, or time travelling, perhaps it is because you are encountering constraints imposed by reality.

Despite empirical evidence, people often [[Does objective reality exist?|argue against the existence of reality]]. In these arguments people may cite the [[w:Allegory_of_the_cave|allegory of the cave]], the [[w:Brain_in_a_vat|brain in a vat]], the [[w:Simulation_hypothesis|simulation hypothesis]], the [[w:The_Matrix_(franchise)|Matrix movies]], claims that [[Facing_Facts/Perceptions_are_Personal|perception is reality]], and [[w:Postmodernism|postmodern theories]].

While these are fascinating thought experiments that do deserve some serious philosophical investigation, they don’t provide much help in getting through our daily lives. I bet that reality exists.&lt;ref&gt; Although I am very confident that reality exists, no one can be [[w:Certainty|certain]]. Much of this course demonstrates the value of embracing and exploring [[w:Doubt|doubt]] while deferring certainty. Also, it is often [[Wisdom|wise]] to think in bets. See: {{cite book |last=Duke |first=Annie |author-link=w:Annie_Duke |date= |title=Thinking in Bets: Making Smarter Decisions When You Don't Have All the Facts |publisher=Portfolio |pages= 288 |isbn=978-0735216372}} &lt;/ref&gt;

For the remainder of the course, we proceed with the assumption that reality exists.

Reality is vast, complex, and dynamic. Humans have only investigated a small portion of the universe, and our investigation is incomplete. Our awareness, observations, and perceptions of reality are neither complete nor accurate representations of reality. We do not observe the many [[w:Cosmic_ray|cosmic rays]] passing through us each second, the atoms that make up the materials we encounter, billions of galaxies beyond the limits of our vision, the ultrasonic chirps used by bats to navigate, viruses, DNA, antibodies, [[w:Greenhouse_gas |greenhouse gasses]], [[w:Particulates|particulate contaminants]], and much more of reality as it is.

This course makes the further assumption that [[Knowing_How_You_Know/One_World|we all live in the same universe]]&lt;ref&gt;This assumption does not conflict with [[w:Many-worlds_interpretation|many-worlds interpretations]] of quantum mechanics. &lt;/ref&gt;. All life forms discovered so far live together on our single planet, circling our sun, in our humble place in the universe. The universe is vast, yet it is all one world, and we all live together on this one planet we call Earth. All that we know of and all that we have ever experienced follow the same laws of physics. Remarkably, the entire world as we know it has emerged from a few [[w:Standard_Model|fundamental building blocks]].

Because we all live in the same universe, our reliable understanding of that universe must eventually converge toward one coherent description. Each phenomenon we observe must fit into a single coherent and integrated description of our universe. Either the description must evolve to accommodate each new observation, or our understanding of that observation must be interpreted consistently with that unified representation.

Building on these assumptions, it follows that [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]], even though any number of [[Exploring Worldviews|worldviews]] is possible. It is likely that each of us hold a worldview that is somewhat different from others. Of all the possible worldviews, one worldview is especially important. That is the worldview that corresponds to reality as closely as our best current understanding of reality allows. Because reality exists, we can [[Evaluating_Evidence|examine reality]], and we can [[Exploring_Worldviews/Aligning_worldviews|align our worldview]] with reality. Because we all live in one world, reality is our common ground.
=== Assignment ===
#Complete the Wikiversity course on [[Facing Facts]].
#Read the essay [[Facing_Facts/Perceptions_are_Personal|Perceptions are personal]].
#Investigate further any of the arguments that appeal to you, listed above, that reality does not exist.
#If you can accept the assumption that reality exists, please proceed with the remainder of this assignment.
#Estimate what fraction of reality you are familiar with.
##Read the essay, [[Virtues/Humility/Being_99.9%_Ignorant|Being 99.9% Ignorant]].
##How many countries are there in the world? How many have you visited?
##Study modern research on the [[w:Galaxy#Modern_research|number of galaxies in the universe]]. How many have you visited?
##Scan this list of [[w:Lists_of_unsolved_problems|lists of unsolved problems]]. What fraction of reality has been investigated by humans?
##Can you accept the premise that reality exists far beyond our perceptions, investigations, and conceptions of it?
#Read the essay [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]].
#Read the essay [[Knowing_How_You_Know/One_World|One World]].
#Read the essay [[Facing_Facts/Reality_is_our_common_ground|Reality is our common ground]].
#Read the essay [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]].
#Optionally read the essay [[Beyond_Theism/What_there_is|What there is]]. Consider writing your own such essay.  
#Can you accept the premise that reality exists, we live in one world, we share one reality, and that reality  is our common ground? If so, please proceed with the remainder of this course.

== Perception ==
[[w:Perception|Perception]] transforms sensory information originating with some material object, known as the ''target'' or the ''[[w:Distal|distal]] stimulus'' into mental representations known as [[w:Perception#Process_and_terminology|percepts]], which do not exist elsewhere in the world. 

[[w:Perception|Perception]]—the process of extracting information from energy that impinges on our [[w:Sense|sensory organs]]—is not straightforward. There is more to perception than meets the eye.&lt;ref&gt;{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }}Page 13 of 162.&lt;/ref&gt;

Indirect theories of perception describe it as a constructive process that involves inference, learning, and experience.&lt;ref&gt;{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }}Page 13 of 162.&lt;/ref&gt; 

While it may seem reasonable that an ideal perception system would duplicate the observed target object exactly in the mental percept, this is not what happens. One theory hypothesizes that the purpose of perception is to allow organisms to locate and use [[w:Affordance|affordances]]—what the environment offers the individual—for practical benefit.

Our most direct encounters with reality are through direct sensory observations and awareness. However, our perceptions are not exact replicas of the distal objects being perceived. Omissions, distortions, and additions take place. Fortunately, we can augment our direct perceptions to obtain a more accurate understanding of our world.

=== Omissions ===
Our direct perception accounts for only a tiny fraction of reality.

We can only directly perceive stimuli that our senses encounter, and these senses experience only a small fraction of reality. We do not see what is behind us, the places we do not visit, or stimuli outside the range of our sensory systems. Attention is selective, so we are likely to perceive only what our attention is drawn to, remaining unaware of the rest. When our attention is captured by moving objects, shiny objects, or unexpected occurrences, we may be distracted from sensing other information in our environment.
[[File:Directing_Attention.jpg|300px|thumb|Our perceptions focus on what we direct our attention toward.]]
Although reality is our common ground, each of us has unique experiences of its vastness. Refer to the diagram on the right. Reality is vast, complex, and dynamic, and the range of experiences any of us has is a tiny fraction of reality as it is. Furthermore, within our range of experience, our attention is limited. Our attention concentrates awareness on some phenomena to the exclusion of other stimuli.&lt;ref&gt;The idea of representing attention as a vertical line sampling reality is presented in {{cite book|title=Liminal Thinking: Create the Change You Want by Changing the Way You Think  |last=Gray|first=Dave|date=September 14, 2016|publisher=Two Waves Books|isbn=978-1933820460|pages=184|author-link=}}&lt;/ref&gt; Because our concentration is selective and focuses on discrete information, whether subjectively or objectively, we exclude other stimuli present even within our range of experience.

Our perceptions focus on what we direct our attention toward.

The spectrum of [[w:Visible_spectrum|light visible to humans]] is only a tiny fraction of the [[w:Electromagnetic_spectrum|electromagnetic spectrum]]. Radio waves, microwave radiation, infrared radiation, visible light, ultraviolet radiation, x-rays, and gamma radiation are all electromagnetic waves of various wavelengths. These various forms of radiation span wavelengths ranging from as small as 10&lt;sup&gt;-11&lt;/sup&gt; meters (0.1 Å) to as long as 1,000 meters, however only light in the range of 400 [[w:Nanometre|nm]] – 700 nm (4x10&lt;sup&gt;-7&lt;/sup&gt; to 7x10&lt;sup&gt;-7&lt;/sup&gt; meters) is visible to humans. We have no way to directly sense radio waves, infrared and ultraviolet light, x-rays, and other forms of radiation. Humans are also unable to directly sense magnetic fields.

The [[w:Hearing_range|range of human hearing]] is approximately 20 to 20,000 [[w:Hertz|Hz]], while dogs can hear sounds frequencies as high as 45 kHz and bats can hear frequencies as high as 200 kHz. Elephants can hear sounds at 14–16 Hz, while some whales can hear [[w:Infrasound|infrasound]] as low as 7 Hz.

It is estimated that dogs, in general, have an [[w:Sense_of_smell#Variability_amongst_vertebrates|olfactory sense]] (sense of smell) approximately ten thousand to a hundred thousand times more acute than a human's.

Also, we are often [[Exploring_Worldviews/What_Fish_Don’t_See|unaware of what could be most obvious]]. 

Not only does our direct perception account for only a tiny fraction of reality, much of what we do perceive is distorted.

=== Distortions: ===
Distortions of the [[w:Sense|senses]] are called [[w:Illusion|illusions]]. Although illusions distort our perception of reality, they are generally shared by most people. Illusions may occur with any of the human senses, but visual illusions (optical illusions) are the best-known and understood.

[[File:Checker shadow illusion.svg|thumb|The squares marked A and B are the same shade of gray.]]
In the [[w:Checker_shadow_illusion|checker shadow illusion]], shown here, two identically colored squares labeled “A” and “B” appear to be very different shades of gray. Reality as we perceive it may not be reality as it is.

[[Image:Fraser spiral.svg|thumb|right|225px|Fraser spiral illusion]]
In the Fraser spiral illusion, shown here, the overlapping black arc segments appear to form a spiral; however, the arcs are a series of concentric circles.

[[File:Duck-Rabbit illusion.jpg|thumb| What do you see? Are you sure?]]
In addition to illusions, many [[w:Ambiguous_image|images are ambiguous]]. As an example, please observe, and then interpret the image shown here.

What do you see? What else do you see? How certain are you of your conclusion? What do you have at stake in defending your interpretation?

Most people will see either a rabbit or a duck. If you see a rabbit, look again, and try to see a duck. Similarly, if you see a duck, look again, and try to see a rabbit.

This image, called the [[w:Rabbit-duck_illusion|rabbit-duck illusion]] is inherently [[w:Ambiguity|ambiguous]]. Someone who interprets it as representing a duck has as valid a claim to accuracy as someone who interprets this as representing a rabbit. In fact, it is neither and both. It is an ambiguous image open to at least two different yet equally valid interpretations. Similarly, a [[w:Is_the_glass_half_empty_or_half_full?|glass that is half empty]] is also half full.

In addition to optical illusions, we are susceptible to illusions that distort each of our senses. These include [[w:Auditory_illusion|auditory]], [[w:Tactile_illusion|tactile]], [[w:Time_perception#Types_of_temporal_illusions|temporal]], and [[w:Illusion#Intersensory|intersensory illusions]].

[[w:Magic_(illusion)|Magicians]] are performing artists who use illusions to entertain us. [[w:Charlatan|Charlatans]], including fakes, [[w:Mysticism|mystics]], [[w:Counterfeit|counterfeiters]],  [[w:Quackery|quacks]], [[w:Conspiracy_theory|conspiracy theorists]], and [[w:Confidence_trick|con artists]], use illusions to deceive, cheat, and swindle us. [[w:Ideology|Ideologs]] exploit illusions to promote their cause, [[w:Special_effect|special effects]] are often used in films and other artistic media to entertain us, and recently [[w:Deepfake|deepfakes]] are used to deceive us. 

In addition to our perceptions being distorted, we also sometimes perceive things which are not there. 

=== Additions: ===
Our perception systems create [[w:Perception#Process_and_terminology|percepts]] that do not otherwise exist in the world. Examples include [[w:Color_vision|color perception]], [[w:Illusory_contours|illusory contours]] and other patterns studied within [[w:Gestalt_psychology|Gestalt psychology]], [[w:Apophenia|illusory pattern recognition]], pain, phantom limbs, [[w:Hallucination|hallucinations]], and distortions that occur during various [[w:Altered_state_of_consciousness|altered states of consciousness]]. Our conceptions also influence our perceptions.

Humans [[w:Perception|perceive]] the [[w:Red|color red]] when we look at light with a wavelength between approximately 625 and 740 [[w:Nanometre|nanometers]]. Note that although electromagnetic radiation with a wavelength of approximately 700nm does exist in objective reality independent of the direct perception of any human, the percept we call red only exists in  our minds as a construct of the perceptual system. This is a consequence of our [[w:Trichromacy|trichromatic color vision]] system, including our visual cortex and [[w:Color_vision#Color_in_the_human_brain|other brain regions]]. Because color only exists as a feature constructed by the visual perception of an observer it is a [[w:Color_vision#Subjectivity_of_color_perception|subjective experience]].  When I see red, I have a subjective experience called redness. However, philosopher John Locke proposed a thought experiment, called the [[w:Inverted_spectrum|inverted spectrum]], where we imagine two people sharing their color vocabulary and discriminations, although the colors one sees—one's [[w:Qualia|qualia]]—are systematically different from the colors the other person sees. For example, perhaps the color we call red creates within me the subjective experience that the color we call green generates within you. We do not know if the subjective experience I call redness is like the subjective experience you have when seeing red.&lt;ref&gt;Siegel, Susanna, &quot;[https://plato.stanford.edu/archives/fall2021/entries/perception-contents The Contents of Perception]&quot;, The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), Edward N. Zalta (ed.), Section 4.1.&lt;/ref&gt;  The nature of redness is further explored in a philosophical thought experiment known as the [[w:Knowledge_argument|knowledge argument]], or Mary’s room.

[[Image:Kanizsa triangle.svg|thumb|right|''Kanizsa's triangle'': Do you see a white triangle?]]

This image, called [[w:Illusory_contours|Kanizsa’s triangle]], gives the impression of a bright white triangle, defined by a sharp illusory contour, occluding three black circles and a black-outlined triangle. Even knowing the bright white triangle does not exist, it is reliably constructed by our visual perception system. This is one example of many [[w:Illusory_contours|illusory contours]] that evoke the perception of an edge without a shading or color change across that edge. We see an edge where none exists. [[w:Gestalt_psychology|Gestalt psychology]] studies many such images where it appears that “The whole is more than the sum of its parts.”

[[w:Apophenia|Apophenia]] is the tendency to perceive meaningful connections between unrelated things. For example Gamblers may imagine that they see patterns in the numbers that appear in lotteries, card games, or roulette wheels, where no such patterns exist.

[[File:Martian face viking cropped.jpg|thumb|Satellite photograph of a [[w:mesa | mesa]] in the [[w:Cydonia (Mars)|Cydonia region of Mars]], often called the [[w:Cydonia (Mars)#&quot;Face on Mars&quot;|&quot;Face on Mars&quot;]] and cited as evidence of [[w:Extraterrestrial life|extraterrestrial]] habitation.]]

One form, called [[w:Pareidolia|pareidolia]], is the tendency for perception to impose a meaningful interpretation on a nebulous stimulus, usually visual, so that one sees an object, pattern, or meaning where there is none. People may mistakenly interpret an object, shape, or configuration with perceived &quot;face-like&quot; features as being a face. Many people claim to see the [[w:Man_in_the_Moon|man in the moon]]. In this satellite photograph of a [[w:mesa | mesa]] in the [[w:Cydonia (Mars)|Cydonia region of Mars]], often called the [[w:Cydonia (Mars)#&quot;Face on Mars&quot;|&quot;Face on Mars&quot;]] has been cited as evidence of [[w:Extraterrestrial life|extraterrestrial]] habitation.

As another example, what we feel as [[w:Pain|pain]] is our perception of tissue damage. Consider our perception of a [[w:Toothache|toothache]]. The damaged tooth stimulates nerve impulses in our [[w:Somatosensory_system|somatosensory nervous system]]. The brain perceives various nerve impulses transmitted through these neural structures as the unpleasant sensory and emotional experience we call pain. Notice that the brain receives only a pattern of nerve impulses and then constructs the perception of pain from those nerve impulses. Although the damage is in the tooth the perception of pain is constructed in the brain.

A [[w:Phantom_limb|phantom limb]] is the sensation that an amputated or missing limb is still attached. Approximately 80 to 100% of individuals with an amputation experience sensations in their amputated limb. In phantom limb syndrome, there is sensory input indicating pain from a part of the body that no longer exists. This phenomenon is still not fully understood, but it is hypothesized that it is caused by activation of the [[w:Somatosensory_cortex|somatosensory cortex]]. The perception of the absent limb is constructed by our nervous system.

As another example, [[w:Tinnitus|tinnitus]] is the perception of sound when no corresponding external sound is present.

Our perceptions are often distorted when we are experiencing [[w:Altered_state_of_consciousness|altered states of consciousness]]. This may occur from the influences of drugs—especially [[w:Psychoactive_drug|psychoactive drugs]] including [[w:Hallucinogen|hallucinogens]] and [[w:Alcohol_(drug)|alcohol]]—fatigue, disease, [[w:Hypoxia_(medical)|hypoxia]] (as can occur from [[w:Hypoventilation|hypoventilation]] and other [[w:Breathing|breath]] control practices), [[w:Hallucination|hallucinations]], [[w:Delirium|delirium]], and various mental illnesses. 

Our conceptions often influence our perceptions. Try this simple quiz:
&lt;blockquote&gt;
What does F-O-L-K spell? (Please say the word out loud.)&lt;br&gt;
What is the white of an egg called?  (Please say the word out loud.)&lt;br&gt;
(What color is the white of an egg?)
&lt;/blockquote&gt;
If you said “yolk”, as many people do, you were influenced by psychological priming. [[w:Priming_(psychology)|Priming]] is a phenomenon whereby exposure to one stimulus influences a response to a subsequent stimulus, without conscious guidance or intention. For example, in experiments the word ''nurse'' is recognized more quickly following the word ''doctor'' than following the word ''bread''.

We may be unaware of priming effects that arise [[w:Priming_(psychology)#In_daily_life|in our daily lives]]. In one study subjects were implicitly primed with words related to the stereotype of elderly people (example: Florida, forgetful, wrinkle). While the words did not explicitly mention speed or slowness, those who were primed with these words walked more slowly upon exiting the testing booth than those who were primed with neutral stimuli.

There is more to perception than meets the eye!

=== Assisted Perception—Compensating for the quirks ===
Fortunately, there are many methods we can use to overcome and compensate for the deficiencies, distortions, quirks, and other characteristics of our direct perception systems. Microscopes, telescopes, [[w:Oscilloscope|oscilloscopes]], many other [[w:Scientific_instrument|scientific instruments]], recording devices, and [[w:Technical_standard|reference standards]] allow us to extend the reach, scope, and accuracy of our observations. Multiple observers, vantage points, perspectives, and viewpoints allow us to gain a more complete and reliable examination of reality when we share and integrate information. 

Applying the principle of [[w:Consilience|consilience]] and [[Thinking Scientifically|thinking scientifically]] increase the reliability of our observations and can help us see beyond illusions. 

==== Assignment ====
#Observe the moon in the sky some evening.
#Estimate the size (diameter) of the moon.
#Estimate the distance the moon is from you.
#Research the [[w:Moon|accepted values of these numbers]] and compare them to your direct observations. How closely does the perceived distance and size compare to the accepted values?
#Optionally repeat this for the sun, bright stars, and other distant terrestrial and celestial objects. 

=== Perceptions are Personal ===
We often hear that “perception is reality” and that “everything is relative”, despite knowing that a shared reality exists, and reality is our common ground.

Perceptions are vivid. Seeing things from our own point of view is always easier, and first-hand experiences seem more real than understanding another's point of view can ever be. Our eyes, nose, taste buds, tactile sensors, and ears connect directly only to our brain. Only you experience first-hand the direct sensory input of the world; you, your self, is the observer. This raw sensory input is interpreted and gains meaning through your unique perceptions and past experiences. Furthermore, contemplation, desire, intent, pain, introspection, consciousness, and reflection are all private and solitary. This unique first-person experience creates a fundamental asymmetry that contributes to many of the other asymmetries that govern social interactions. It also contributes to the asymmetric character of [[Coping with Ego|egotism]], narcissism, selfishness, greed, and the magnitude gap. We judge others based on behavior and we judge ourselves based on intent. Your own point of view, the way you see things, is unique. The [[Living the Golden Rule|golden rule]] and our empathy struggle to overcome this fundamental imbalance.

It is often a [[w:Problem_of_induction|mistake to generalize]] our personal perceptions beyond our own experiences. Standing in a meadow we see a flat earth, yet sunrise, time zones, global travel, earth satellites, GPS navigation systems, images from space, and travel to the moon all assure us the [[w:Spherical_Earth|earth is nearly spherical]]. Reality is vast, complex, and dynamic, and our perceptions are only a tiny glimpse of all there is to know about reality. Only a [[Global Perspective|global perspective]] brings us uncensored reality.

Reality exists and provides us with [[Facing_Facts|matters of fact]]. The [[Knowing_How_You_Know/Height_of_the_Eiffel_Tower|Eiffel tower is 300 meters tall]]. You may perceive that as too short; others may perceive that as too tall, and many perceive that as simply beautiful.
 
See beyond the illusion that what you see is all there is. Perceptions are personal, but [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]]. Our unique first-person viewpoint creates a powerful asymmetry that requires deliberate effort to see beyond. Transcend your personal perceptions, investigate the vastness and complexity of reality from a [[Global Perspective|global perspective]], and embrace reality as our common ground.
==== Assignment ====
#Read the essay [[Facing_Facts/Perceptions_are_Personal|Perceptions are Personal]].
#Reflect on occasions when you projected your own perceptions onto others or beyond your own experience.
#Recognize the limited scope and likely deficiencies of your perceptions.

== Representation ==
At least until we become proficient at [[w:Telepathy|mental telepathy]], we need some other way to represent our thoughts and to communicate our thoughts to others.

[[w:Word|Words]] are one type of [[w:Symbol|symbol]] used to represent reality. Other symbols might include [[w:Vocable|sounds]], [[w:List_of_gestures|gestures]], [[w:Facial_expression|facial expressions]], [[w:Icon_(computing)|icons]], [[w:Logo|logos]], [[w:Drawing|drawings]], [[w:Model|models]], [[w:Analogy|analogies]], [[w:Metaphor|metaphors]], and other representations. The mapping from words to perceptions to reality [[w:Polysemy|is often imprecise]]. We may describe a certain color simply as ''red'', without clarifying what [[w:Shades_of_red|shade of red]] we are describing. If I tell you I am sitting in a chair, you might imagine a rocking chair, a desk chair, or a recliner. [[w:Syntactic_ambiguity|Syntactic ambiguity]] often adds to the confusion.

Do not mistake the symbol for reality. The [[w:Noumenon|noumenal]] world exists independent of our senses and may differ from the [[w:Noumenon|phenomenal]] world we perceive and the symbols we use to represent it. 

Language and other symbols can be constraining, ambiguous, refer to social constructs, prejudiced, based on artificial boundaries, or focus on convenient representation rather than reality. These features, distortions, and inaccuracies are explored further below.

=== Constraining Language ===
[[File:Clip.jpg| thumb|right|What can you do with this wire form? What would you call it?]]
Consider the image on the right. What can we use it for? This wire form is commonly called a [[w:Paper_clip|paper clip]] and is typically used to clip together sheets of paper. However, [[Thinking_Tools#Lateral_Thinking|lateral thinkers]] have identified at least one [[w:Paper_clip#Other_uses|hundred alternative]] uses&lt;ref&gt;100 Uses for Paperclips, See: https://leoniehallatinnovationiq.wordpress.com/2012/11/21/100-uses-for-paperclips/ &lt;/ref&gt;  for this object. Observing the object, thinking expansively and creatively about what it might be, how it can be used, or what it might do, can identify many possibilities. This unclassified reality invites many potential uses. Once the object is named, an interpretation is imposed, and the object becomes restricted.

Naming the object assigns the object to one category, as it excludes others. 

This table describes possibilities before and after it is interpreted, assigned to a category, and named.

{| class=&quot;wikitable&quot;
|+ Categorization Constrains Imagination
|-
! Encounter with reality: !! Interpretation and Representation:
|-
|
*An experience
*Awareness
*Curious
*Exploration
*Investigation
*Witness
*Imagine
*Possibilities
*Potential
*Opportunities
*An affordance
*Lateral Thinking
*No (one) thing
*The territory
*Could be …
||
*Analogy
*Categories
*Labels
*Restrictions
*Evaluation
*Judgement
*Purpose
*Specifics
*''This'' map
*''This'' thing
**A paperclip
**A lock pick
**A hook
**DVD drive opener
**Earrings
|}
	
Language and other symbols are ambiguous, persuasive, and subtle, and can be used heroically, precisely, expansively, restrictively, deceptively, and manipulatively. Here are some examples.

=== Ambiguous Language ===
[[w:Polysemy|'''Polysemy''']] is the capacity for a sign (e.g., a symbol, a word, or a phrase) to have multiple related meanings. A word can have several [[w:Polysemy|word senses]]. As an example, the word ''bank'' can have at least these 7 distinct meanings:
#a financial institution
#the physical building where a financial institution offers services
#to deposit money or have an account in a bank (e.g., &quot;I bank at the local credit union&quot;)
#[[w:Bank_(geography)|a steep slope]] (as of a hill or the rising ground bordering water)
#[[w:Banked_turn#Banked_turn_in_aeronautics|to incline an airplane]] laterally
#a supply of something held in reserve: such as &quot;banking&quot; brownie points
#a synonym for 'rely upon' (e.g. &quot;I'm your friend, you can bank on me&quot;).

[[w:Syntactic_ambiguity|'''Syntactic ambiguity''']] is language where a [[w:Sentence_(linguistics)|sentence]] may be interpreted in more than one way due to ambiguous [[w:Syntax|sentence structure]]. For example, the sentence ''John saw the man on the mountain with a telescope'' can have these various interpretations. 
*John, using a telescope, saw a man on a mountain.
*John saw a man on a mountain which had a telescope on it.
*John saw a man on a mountain who had a telescope.
*John, on a mountain and using a telescope, saw a man.
*John, on a mountain, saw a man who had a telescope.

Combining polysemy with syntactic ambiguity results in multiplying the ambiguity. For example, Gerald Weinberg identifies dozens of interpretations of the sentence “Mary had a little lamb”&lt;ref&gt; {{cite book |last1=Gause  |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is  |publisher=Dorset House Publishing Company |pages=176 |isbn=978-0932633163 |author-link=w:Gerald_Weinberg}} @128 of 255.&lt;/ref&gt;  and then goes on to suggest 20 more techniques for identifying ambiguity in sentences.

[[Exploring_Social_Constructs#Reifications|'''Reifications''']] are a class of nouns that do not refer to real objects but rather to abstract concepts. Examples include justice, freedom, liberty, equality, rights, duty, responsibility, truth, fairness, and citizen and extend to include concepts such as race, land ownership, owning coal, money, debt, contracts, and even agreement. These labels cannot be resolved to [[w:Brute_fact#Searle|brute facts]] but are often treated as if they do. This results in a [[w:Reification_(fallacy)|reification fallacy]]. A [[w:Map–territory_relation#&quot;A_map_is_not_the_territory&quot;|map is not the territory]], an [[w:The_Treachery_of_Images|image of a pipe is not a pipe]], and there is little or no agreement on what “[[w:justice|justice]]” is. 

Simple phrases such as “we seek justice”, “justice was served”, “[[w:Pledge_of_Allegiance|liberty and justice for all]]”, “[[w:Preamble_to_the_United_States_Constitution|establish justice]]”, and “ours is a nation of laws” become very ambiguous, complex, and controversial when we recognize the wide range of meanings attributed to the abstract concept of “justice”.

The concept of [[w:justice|justice]] differs in every culture. Early theories of justice were set out by the Ancient Greek philosophers Plato in his work [[w:Republic_(Plato)|The Republic]], and Aristotle in his [[w:Nicomachean_Ethics|Nicomachean Ethics]]. Throughout history various theories have been established. Advocates of [[w:Divine_command_theory|divine command theory]] argue that justice issues from God. In the 1600s, theorists like [[w:John_Locke|John Locke]] argued for the theory of [[w:Natural_law|natural law]]. Thinkers in the [[w:Social_contract|social contract]] tradition argued that justice is derived from the mutual agreement of everyone concerned. In the 1800s, [[w:Utilitarian|utilitarian]] thinkers including [[w:John_Stuart_Mill|John Stuart Mill]] argued that justice is what has the best consequences. Theories of [[w:Distributive_justice|distributive justice]] concern what is distributed, between whom assets or liabilities are to be distributed, and what is the ''proper'' distribution. [[w:Egalitarianism|Egalitarians]] argued that justice can only exist within the coordinates of equality. [[w:John_Rawls|John Rawls]] used a [[w:Social_contract|social contract]] argument to show that justice, and especially distributive justice, is a form of [[Understanding Fairness|fairness]]. Property rights theorists (like [[w:Robert_Nozick|Robert Nozick]]) also take a consequentialist view of distributive justice and argue that property rights-based justice maximizes the overall wealth of an economic system. Theories of retributive justice are concerned with [[w:Punishment|punishment]] for wrongdoing. [[w:Restorative_justice|Restorative justice]] (also sometimes called &quot;reparative justice&quot;) is an approach to justice that focuses on the needs of victims and offenders.

There are many [[w:Wikipedia:Manual_of_Style/Words_to_watch|words to avoid]] when trying to be objective and precise. 
===== Assignment =====
#Choose a speech or other text to use for this assignment. This may be chosen from this [[w:List_of_speeches|list of speeches]], the Wikisource [[s:Portal:Speeches|speeches portal]], [[w:List_of_amendments_to_the_United_States_Constitution|amendments to the United States Constitution]], the [[w:Universal_Declaration_of_Human_Rights|Universal Declaration of Human Rights]], [[w:List_of_landmark_court_decisions_in_the_United_States|landmark court decisions]], or any other source.
#Identify instances of ambiguous language in the chosen text.
#For at least three instances of ambiguous language, list various meanings that can be reasonably implied by the language used. For example, the ambiguous phrase “in the future” could mean seconds, hours, days, weeks, months, years, centuries, or eons from now. 
#Suggest more precise and objective language that could be substituted for the language used. 
#Suggest an [[w:Operational_definition|operational definition]] to replace or clarify the ambiguous language. 
==== Social Constructs ====
When we endow [[w:Brute_fact#Searle|''brute facts'']] with additional status, we create [[Exploring_Social_Constructs|''social constructs'']].

Social constructs rely on collective human agreement, or at least acceptance of the imposition of the new status Y on the brute fact X.

Social constructs include games such as soccer, baseball, and chess. Bureaucracies including clubs, organizations, and corporations are social constructs. Titles such as chairman, president, king, and pope, along with governments, financial instruments, property ownership agreements, and religions are social constructs.

Social constructs are [[Exploring_Social_Constructs#Ambiguity|ambiguous]], sometimes fragile, and often require [[Exploring_Social_Constructs#Referees|referees]] of some form. The agreements used to form the social constructs may be obsolete or challenged. Social constructs can be [[Exploring_Social_Constructs#Mismatches|mismatched]] to relevant brute facts.

Because social constructs are so common and so prominent, we can easily mistake them for brute facts. This is an error. Mistaking social constructs for brute facts introduces several layers of abstraction and creates distortions that distance us from realty. 

As a result of ambiguity and defective agreements many social constructs are poorly aligned with the brute facts they are based on, and many mismatches occur. These cause friction in our society and can contribute to [[Grand challenges|many challenges]] we face. By observing the mismatch of social constructs to brute facts and informed consent, we can begin to troubleshoot and improve the collection of social constructs that create our culture and institutions.

The bad news is that we face many issues resulting from social constructs misaligned with brute facts or based on defective agreements. The good news is that because social constructs are human constructs, we can work to improve them.
===== Assignment =====
#Choose a speech or other text to use for this assignment. This may be chosen from this [[w:List_of_speeches|list of speeches]], the Wikisource [[s:Portal:Speeches|speeches portal]], [[w:List_of_amendments_to_the_United_States_Constitution|amendments to the United States Constitution]], the [[w:Universal_Declaration_of_Human_Rights|Universal Declaration of Human Rights]], [[w:List_of_landmark_court_decisions_in_the_United_States|landmark court decisions]], or any other source.
#Identify instances of ''social constructs'' that appear in the chosen text.
#For at least three instances of the social constructs identified, list various meanings that can be reasonably implied by the language used. For example, the [[w:Second_Amendment_to_the_United_States_Constitution|second amendment]] uses the term “[[w:Militia|Militia]]”.  This could mean military unit, police units, paramilitary units, security guards, or individuals. 
#Suggest more precise and objective language that could be substituted for the language used. 
#Suggest an [[w:Operational_definition|operational definition]] to replace or clarify the ambiguous language. 

==== Prejudiced Language ====
[[w:Loaded_language|Loaded language]] is [[w:Rhetoric|rhetoric]] used to influence an audience by using words and phrases with strong [[w:Connotations|connotations]]. This type of language is unclear (vague) and can be used to [[w:Pathos|invoke an emotional response]] or exploit [[w:Stereotypes|stereotypes]].  Loaded words and phrases have significant emotional implications and involve strongly positive or negative reactions beyond their [[w:Literal_meaning|literal meaning]]. These may include [[w:Racism|racists]] or [[w:Sexism|sexist]] words, [[w:List_of_ethnic_slurs|ethnic slurs]], [[w:List_of_religious_slurs|religious slurs]], and other [[w:Lists_of_pejorative_terms_for_people|pejorative terms]]. 

[[w:Wikipedia:Manual_of_Style/Words_to_watch#Contentious_labels|Examples of contentious labels include]]: cult, racist, perverted, sexist, homophobic, transphobic, misogynistic, sect, fundamentalist, heretic, extremist, denialist, terrorist, freedom fighter, bigot, myth, neo-Nazi, -gate, pseudo-, controversial, and others.
==== Beware of Boundaries ====
[[File:Tannin heap.jpeg|thumb|The [[w:Sorites_paradox|sorites paradox]]: If a heap is reduced by a single grain at a time, at what exact point does it cease to be considered a heap?]]

The [[w:Sorites_paradox|sorites paradox]] poses the question, if removing one grain from a heap of sand leaves it a heap, then one grain of sand is also a heap. When does a heap of sand transform into a few grains of sand that are no longer a heap? This paradox illustrates that the concept of ''heap'' is ambiguous. Also, the boundary between a heap and some smaller collection is also ambiguous. 

In conventional language, logic, mathematics, and decision-making, we generally regard [[Natural_Inclusion/Boundaries|boundaries]] as discrete or definitive limits or borders, which permanently and absolutely divide one thing or locality apart from other things or localities. For such definitive limits to exist, however, they would have to be so sharp as to have no thickness. By contrast, even when viewed from afar and over short durations, natural boundaries often appear diffuse, mobile, and impermanent, defying such precise, abstract definition. Naturally occurring boundaries are inherently ambiguous. Artificial boundaries are often sharply defined. This mismatch between how boundaries occur naturally and how we chose to represent them can lead to several problems. 

Racial classifications create problems because they impose sharply defined boundaries where no natural boundary exists. [[w:Race_(human_categorization)|Racial classifications]] are [[Exploring_Social_Constructs|socially constructed]]. While partially based on physical similarities within groups, race does not have an inherent physical or biological meaning. Therefore, race assignment is inherently ambiguous. None-the less, laws prevail that impose harsh burdens based on racial classification. These laws define sharp boundaries to separate one race from another. Simplistic resolutions of racial ambiguity are the [[w:One-drop_rule|one-drop rules]] that asserted that any person with even one ancestor of black ancestry ('one drop' of 'black blood') is considered black. Attempts to define [[w:Native_American_identity_in_the_United_States#Blood_quantum|Native American identity in the United States]] encounters similar difficulties.

==== The map is not the territory. ==== 
As soon as we label an object to represent it, use a [[w:Mental_model|mental model]], invoke an [[w:Analogy|analogy]], use a [[w:Metaphor|metaphor]], substitute a representation for a thought, idea, or object, or substitute an interpretation for some set of observations we substitute [[w:Map–territory_relation|the map for the territory]].

Our brain learns a model of the world. Intelligence is tied to the model the brain creates. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}@22 of 384&lt;/ref&gt;  The brain learns its model of the world by observing how its inputs change over time. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}} @66 of 384&lt;/ref&gt; The neocortex learns a predictive model of the world. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}@75 of 384&lt;/ref&gt;

We rely on the map our brain creates to navigate the world we live in. This works best when our mental maps correspond accurately to the real world. We navigate through life guided by the mental representations that form in our brains. If you are now sitting on a chair in a room, and you want to leave that room, you rely on your mental image of the chair, your location in the room, and the location of the door. If this mental model is wrong, you will have difficulty getting out of the chair, walking across the room, opening the door, and walking through.

We know [[w:Map–territory relation|the map is not the territory]], but merely some simplified and often distorted representation of the corresponding territory—reality as it is. Whenever we rely on the map rather than the territory for forming beliefs, deciding, or planning actions, we risk misrepresenting the territory and embracing a distorted view of reality.

George Box reminds us that “[[w:All_models_are_wrong|All models are wrong]], some are useful”.

We depart from reality the moment we move from [[w:Perception|''perception'']] to [[w:Concept|''conception'']] from [[w:Observation|''observation'']] to [[w:Interpretation_(philosophy)|''interpretation'']].  In his painting [[w:The_Treachery_of_Images|''The Treachery of Images'']], [[w:René_Magritte|René Magritte]] reminds us that an image of a pipe is a representation of the pipe and not the pipe itself.

It is difficult to be precise and neutral in our language. Language provides many opportunities to be vague, ambiguous, [[w:Bullshit|nonsensical]], prejudicial, emotional, laudatory, persuasive, kind, cruel, [[w:Buzzword|vacuous]], or [[w:Weasel_word|equivocal]]. Because [[w:Rhetoric|rhetoric]] is the art of persuasion, be aware that it can draw you toward a conclusion and influence your beliefs using only baseless arguments and emotional manipulation. 

=== Assignment ===
#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of language, perception, or reality.
#[[Practicing Dialogue|Practice dialogue]] rather than debate or argumentation. Be candid. [[Living_Wisely/Advance_no_falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course.
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[Evaluating_Evidence|research them]].
##[[Seeking True Beliefs|Seek true beliefs]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Interpretation ==
People [[w:Interpretation_(philosophy)|interpret]] observations to resolve ambiguity, increase certainty, and to account for observations within some familiar model or analogy. The role of interpretation is most evident when the items being interpreted are most [[w:Ambiguity|ambiguous]]. 
=== Tea Leaves ===
[[File:Spring pouchong tea leaves on plate.jpg|thumb|''Spring Pouchong'' tea leaves that may be used for [[w:Tasseography|tasseography]] divination]]
[[w:Tasseography|Tasseography]] is the art of interpreting patterns in tea leaves, coffee grounds, or wine sediments. The diviner—a person skilled in interpreting tea leaves—looks at the pattern of tea leaves in the cup and allows the imagination to play around with the shapes suggested by them. They might look like a letter, a heart shape, a ring, or anything else. These shapes are then interpreted intuitively or by means of some system of symbolism. Images formed in a cup are created and uniquely seen by the reader, so it is often said that the only limitation for cup reading is the imagination of the reader themselves.

Tasseography reveals the essence of interpretation at an extreme. Because the tea leaf formations are entirely arbitrary the interpreter can say almost anything. The interpretation reflects judgements, evaluations, biases, concepts, models, analogies, predictions, optimism, pessimism, certainty, risk, doubt, hope, fears, storytelling skills, and desires of the interpreter, entirely independent of the tea leaf formation.

[[w:Rorschach_test|Rorschach tests]], [[w:Astrology|astrology]], [[w:Horoscope|horoscopes]], [[w:Biorhythm_(pseudoscience)|biorhythms]], [[w:Tarot_card_reading|tarot card reading]], [[w:Ouija|Ouija]], [[w:Fortune-telling|fortune telling]], and [[w:Dream_interpretation|dream interpretations]] are similar to tasseography in that they all depend more on the person doing the interpretation than on the ambiguous items being interpreted.

Recognizing the extent of ambiguity inherent in the various representations we use to describe reality, interpretation plays a significant role in influencing our understanding and beliefs. Adopting an interpretation can obscure and nearly occlude the underlying reality. Keep your eye on the territory as others offer various maps they would like you to use instead. Continue to reference reality so you can evaluate, challenge, and often reject, various interpretations.

=== Reality, Perception, and Interpretation ===
[[File:Blind monks examining an elephant.jpg|thumb|A depiction of the [[w:Blind_men_and_an_elephant|blind men and the elephant]]]]
The parable of the [[w:Blind_men_and_an_elephant|blind men and the elephant]] provides an example that can help us examine the distinctions among [[w:Reality|''reality'']], [[w:Perception|''perception'']], and [[w:Interpretation_(philosophy)|''interpretation'']]. In the parable is a story of a group of blind men who have never come across an elephant before and who learn and imagine what the elephant is like by touching it. Each blind man feels a different part of the elephant's body, but only one part, such as the tail or the tusk. They then describe the elephant based on their limited experience and their descriptions of the elephant are different from each other.

The elephant is an example of ''reality''—what exists in the world. Each of the blind men makes some limited ''observations'' and forms a ''perception'' of the elephant. Each of these perceptions is based a small sample of reality. The man at the tail accurately perceives the tail by sensing how it feels. The man at the tusk accurately perceives the tusk, also by sensing how it feels. Then each man ''interprets'' his (limited) observation. The man at the tail concludes an elephant is like a rope, the man at the tusk concludes the elephant is hard, smooth, and like a spear.

Each blind man makes errors when interpreting their perceptions:
*Each fails to include the observation of the others, and if they consult the other observers, they fail to trust these additional observations and integrate them into a consistent whole. 
*Each [[w:Faulty_generalization|overgeneralizes]] from their limited experience to draw conclusions about the entire elephant.
*Each interprets their observations in terms of existing [[w:Paradigm|paradigms]] (rope, spear) rather than considering new paradigms (elephant). 

Recall the differing yet equally correct interpretations of the [[w:Rabbit–duck illusion|rabbit-duck illusion]].  This simple image allows for two different, yet equally correct interpretations. Life is often more complicated than ducks and rabbits, and we can reasonably differ in interpreting many words.

The [[w:Second_Amendment_to_the_United_States_Constitution|second amendment to the United States Constitution]] is the subject of long and contentious interpretations. It states:
&lt;blockquote&gt;
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
&lt;/blockquote&gt;
The (sometimes omitted) commas in the text draw much attention. The Militia clause is especially open to interpretation. Indeed, most of the words in this short statement have been interpreted in a variety of ways. Interpretation of this text is the subject of [[w:List_of_firearm_court_cases_in_the_United_States|many court cases]] and continues to sharply divide political discourse in the United States.

[[w:Innuendo|Innuendo]] and [[w:Plausible_deniability|plausible deniability]] [[w:Deception|disingenuously]] [[w:Equivocation|equivocate]] on interpretation.

Learn to [[Embracing_Ambiguity|embrace ambiguity]]. [[Finding_Common_Ground/Doubt_and_our_Bayesian_Brains|Become comfortable with doubt]]. Examine and reevaluate your preconceptions. Avoid tragic misinterpretations.

=== Tragic Interpretations ===
Interpretations that prematurely eliminate [[w:Doubt|doubt]] and resolve [[w:Ambiguity|ambiguity]] into a comfortable yet false feeling of [[w:Certainty|certainty]] have led to several tragedies. Here are some prominent examples.
#It has long been observed that the sky brightens each morning and day turn to darkness each evening. The causes of this were ambiguous for most of recorded history. Perhaps the sun moves around the earth on a [[w:Celestial_sphere|celestial sphere]]. Alternatively, the earth could rotate on its axis as it [[w:Heliocentrism|revolves around the sun]]. The pope was certain the earth was the center of the universe, [[w:Galileo_Galilei|Galileo]] differed. The contentious [[w:Galileo_affair|debate over these alternative interpretations]] of the observations culminated with the trial and condemnation of [[w:Galileo_Galilei|Galileo Galilei]] by the [[w:Roman_Inquisition|Roman Catholic Inquisition]] in 1633.
#In 1997 members of the [[w:Heaven's_Gate_(religious_group)#Mass_suicide|Heaven’s Gate]] new religious movement misinterpreted images of the Hale-Bopp comet, and decided that the only way to evacuate this earth was to participate in a mass suicide. [[w:Cult|Cults]] are often formed based on alternative interpretations of events.
#Although the phrase “[[w:All_men_are_created_equal|All men are created equal]]” motivated [[w:American_Revolutionary_War|the revolution]] that formed the United States, differing interpretations of the status of [[w:Slavery_in_the_United_States|slaves]] as being either property or being humans led to the [[w:American_Civil_War|civil war]]. 
#Religious groups often differ in the interpretation of various symbols, texts, and prophesies. Here are some examples.
##[[w:Christianity|Christianity]] is a religion based on interpretations of the [[w:Life_of_Jesus_in_the_New_Testament|life]] and [[w:Teachings_of_Jesus|teachings]] of [[w:Jesus|Jesus of Nazareth]]. These interpretations lead to the belief that [[w:Jesus|Jesus]] is the [[w:Son_of_God_(Christianity)|Son of God]], whose coming as the [[w:Messiah#Christianity|messiah]] was [[w:Old_Testament_messianic_prophecies_quoted_in_the_New_Testament|prophesied]] in the [[w:Hebrew_Bible|Hebrew Bible]] and chronicled in the [[w:New_Testament|New Testament]].
##The [[w:Judaism#Christianity_and_Judaism|differences between Christianity and Judaism]] originally centered on whether Jesus was the Jewish Messiah but eventually became irreconcilable. Followers of [[w:Judaism|Judaism]] interpret the life of Jesus to be that of a well-meaning and charismatic human who worked as carpenter, but not the Messiah. This difference in interpretation led to the [[w:The_Holocaust|holocaust]]. 
##[[w:Islam|Islam]] is a religion teaching that [[w:Muhammad|Muhammad is a messenger of God]]. The primary scriptures of Islam are the [[w:Quran|Quran]], interpreted as the verbatim word of God. [[w:Islam#Denominations|Various denominations]] differ in their interpretations of the rightful successors of Muhammad. This difference in interpretation has led to [[w:Human_rights_in_post-invasion_Iraq#Sectarian_warfare_in_Iraq|sectarian warfare]]. 
##[[w:Scientology|Scientology]], [[w:Scientology#Scientology_as_a_religion|classified as a religion]] by the United States Internal Revenue Service, is a set of beliefs and practices invented by American author [[w:L._Ron_Hubbard|L. Ron Hubbard]], and an associated movement. It has been variously defined as a [[w:Cult|cult]], a [[w:Scientology_as_a_business|business]], or a [[w:New_religious_movement|new religious movement]], depending on various interpretations of the various [[w:Scientology_beliefs_and_practices|beliefs and practices]].
##[[w:Nontheism|Nontheists]] study reality and recognize that [[Beyond_Theism#Non-Theism_is_the_Null_Hypothesis|non-theism is the parsimonious worldview]]. Therefore, theists who make supernatural claims bear the (unmet) burden of proving their supernatural claims. Nontheists may risk persecution for [[w:Blasphemy|blasphemy]]. 
##[[w:Religious_war|Religious wars]], often resulting from disputes over these various interpretations, are frequent, long lasting, and deadly. Matthew White's [[w:The_Great_Big_Book_of_Horrible_Things|''The Great Big Book of Horrible Things'']] gives religion as the primary cause of 11 of the world's 100 deadliest atrocities.
#The nature of the [[w:2021_United_States_Capitol_attack|2021 United States Capitol attack]] has been widely, passionately, and [[w:Domestic_reactions_to_the_2021_United_States_Capitol_attack|variously interpreted]]. Former attorney general [[w:William_Barr|William Barr]], who had resigned days earlier, denounced the violence, calling it &quot;outrageous and despicable&quot;, adding that the president's actions were a &quot;betrayal of his office and supporters&quot; and that &quot;orchestrating a mob to pressure Congress is inexcusable.&quot; None-the-less, the Republican National Committee contended that the lethal riot was an example of &quot;legitimate political discourse.&quot; [[w:2021_United_States_Capitol_attack#Aftermath|The aftermath]] continues to have important political, legal, and social repercussions.
#Various [[w:Conspiracy_theory|conspiracy theories]] are based on alternative interpretations of events. Here are a few selected from a much longer [[w:List_of_conspiracy_theories|list of conspiracy theories]].
##Many [[w:John_F._Kennedy_assassination_conspiracy_theories|conspiracy theories concerning the assassination of John F. Kennedy]] in 1963 have emerged. Many of these depend on various interpretations of the available evidence and are especially skeptical of the [[w:Single-bullet_theory|single bullet theory]]—the official description of the event appearing in the [[w:Warren_Commission|Warren commission report]]. It is also frequently asserted that the United States federal government intentionally covered up crucial information in the aftermath of the assassination to prevent the conspiracy from being discovered.
##The [[w:September_11_attacks|multiple attacks]] made on the US by [[w:Terrorism|terrorists]] using hijacked aircraft on September 11, 2001 have proven [[w:9/11 conspiracy theories|attractive to conspiracy theorists]]. Theories may include reference to missile or hologram technology. By far, the most common theory is that the attacks were in fact controlled demolitions, a theory which has been rejected by the engineering profession and the [[w:9/11_Commission|9/11 Commission]].
##The &quot;[[w:Deep_state_in_the_United_States|Deep state]]&quot; often refers to discredited allegations of an unidentified &quot;powerful elite&quot; who act in coordinated manipulation of a nation's politics and government. Proponents of such theories have included Canadian author [[w:Peter_Dale_Scott|Peter Dale Scott]], who has promoted the idea in the US since at least the 1990s, as well as [[w:Breitbart_News|''Breitbart News'']], [[w:Infowars|''Infowars'']] and former US President [[w:List_of_conspiracy_theories_promoted_by_Donald_Trump|Donald Trump]]. A 2017 poll by [[w:ABC_News|ABC News]] and The Washington Post indicated that 48% of Americans believe in the existence of a conspiratorial &quot;deep state&quot; in the US. Some of these theories promote [[w:QAnon|QAnon conspiracy theories]] which are based on the interpretation of false claims made by an anonymous individual or individuals known as &quot;Q&quot;.
##[[w:Anti-vaccination|Anti-vaccination activists]] and other people in many countries have spread a variety of unfounded [[w:COVID-19_vaccine_misinformation_and_hesitancy|conspiracy theories]] and other [[w:Misinformation|misinformation]] about [[w:COVID-19_vaccine|COVID-19 vaccines]] based on misinterpreted or misrepresented science, religion, exaggerated claims about side effects, a story about COVID-19 being spread by [[w:COVID-19_misinformation#5G_mobile-phone_networks|5G]], misrepresentations about how the immune system works and when and how COVID-19 vaccines are made, and other false or distorted information. This has prolonged the pandemic and caused political unrest. 
#[[w:Quackery|Quackery]], [[w:Quackery|crystal healing]], [[w:Homeopathy|homeopathy]], and other ineffective and fraudulent health claims waste time and money while deceiving patients and delaying effective treatments. These are sustained by inaccurate interpretation of evidence.
=== Assignment ===
#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of interpretations, language, perception, or reality.
##Identify the specific interpretations that differ.
##Identify the ambiguities that allow for various interpretations.
##Cast doubt on the certainty of any specific interpretation. 
#[[Practicing Dialogue|Practice dialogue]] rather than debate or argumentation. Be [[Candor|candid]]. [[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course.
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[w:Evaluating_Evidence|research them]].
##[[Embracing Ambiguity|Embrace ambiguity]].
##[[Seeking_True_Beliefs|Seek true beliefs]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Narration ==
[[File:John Everett Millais (1829-1896) - The Boyhood of Raleigh - N01691 - National Gallery.jpg|thumb|We are captivated by [[w:Storytelling|storytelling]].]]
Humans enjoy telling and retelling [[w:Narrative|stories]]. [[w:Myth|Myths]] have been a part of human culture for at least as long as recorded history. [[w:Folklore|Folklore]], [[w:Oral_tradition|oral traditions]], [[w:Epic_poetry|epics]], [[w:Creation_myth|creation myths]], [[w:Campfire_story|campfire stories]], [[w:Fairy_tale|fairy tales]], [[w:Legend|legends]], [[w:Bedtime_story|bedtime stories]], [[w:Song|songs]], [[w:Poetry|poems]], and [[w:Soap_opera|soap operas]] are told and retold. Stories provide a memorable, coherent, compelling, and plausible explanation for events. They are also often fanciful and factually unfounded.

In his book [[w:Sapiens:_A_Brief_History_of_Humankind|''Sapiens: A Brief History Of Humankind'']], author Yuval Harari claims that all large-scale human cooperation systems – including [[w:Religions|religions]], [[w:Political_structures|political structures]], [[w:Trade|trade networks]], and [[w:Legal_institutions|legal institutions]] – owe their emergence to Sapiens' distinctive cognitive capacity for [[w:Fiction|fiction]]. Ours is the storytelling species.

=== Narratives that stick ===
Why is it that some ideas survive while others die? According to the book [[w:Made_to_Stick|''Made to Stick'']], an idea becomes memorable or interesting when it is: 
*Simple – find the core of any idea or thoughts
*Unexpected – grab people's attention by surprising them
*Concrete – make sure an idea can be grasped and remembered later
*Credible – give an idea believability and credibility
*Emotional – help people see the importance of an idea
*Presented as a story – empower people to use an idea through narrative

Several grand narratives currently divide our [[Exploring Worldviews|worldviews]] and political discourse in the United States. [[w:Conservatism|Conservatives]] often agree with Ronald Regan that “[[w:Ronald_Reagan#First_Inaugural_address_(1981)|Government is the Problem]]”, while [[w:Progressivism|progressives]] tend to believe that “Government is the solution”. Each [[w:Tribe|tribe]] can cite many examples bolstering their position. Beginning with their chosen narrative, conservatives often [[w:Mask_refusal|oppose mask mandates]], while progressives blame the unmasked for endangering others and prolonging the pandemic. These arguments begin with narratives and extend to interpretations and representative symbols but are rarely based on a careful [[Evaluating_Evidence|evaluation of evidence]].

Americans are divided by several such narratives. More [[w:Gun_politics_in_the_United_States|guns make us safer]], or they cause needless [[w:Gun_violence_in_the_United_States|violence]]. [[w:Climate_change|Climate change]] is the biggest threat to our future or is simply a hoax. [[w:Abortion_in_the_United_States|Abortion]] murders babies or protects a women’s constitutional right to choose. God created man, or [[w:Problem_of_the_creator_of_God|man created God]]. [[w:Psalm_115|Earth belongs to man]] or [[q:Chief_Seattle|man belongs to earth]]. Capitalism is the solution or [[w:Criticism_of_capitalism|capitalism is the problem]]. Do you believe the experts or do you [[w:Knowing_How_You_Know/Divided_by_epistemology|believe your friends]]? Various conspiracy theories provide especially troublesome narratives. Ideologies amplify [[w:List_of_cognitive_biases|cognitive biases]].
=== Powerful False Narratives ===
Very often, [[w:Storytelling|the best story wins]]. Here is an example of a powerful, influential, and harmful false narrative, known as the ''satanic panic''.

The [[w:Satanic_panic|Satanic panic]] is a [[w:Moral_panic|moral panic]] consisting of over 12,000 unsubstantiated cases of Satanic ritual abuse (SRA) starting in the United States in the 1980s, spreading throughout many parts of the world by the late 1990s, and persisting today. The panic originated in 1980 with the publication of [[w:Michelle_Remembers|''Michelle Remembers'']], a bestselling book co-written by Canadian psychiatrist [[w:Lawrence_Pazder|Lawrence Pazder]] and his patient (and future wife), Michelle Smith, which used the discredited practice of [[w:Recovered-memory_therapy|recovered-memory therapy]] to make sweeping lurid claims about satanic ritual abuse involving Smith. The allegations which afterwards arose throughout much of the United States involved reports of [[w:Physical_abuse|physical]] and [[w:Sexual_abuse|sexual abuse]] of people in the context of [[w:Occult|occult]] or [[w:Theistic_Satanism|Satanic]] rituals. In its most extreme form, allegations involve a conspiracy of a global Satanic cult that includes the wealthy and powerful world elite in which children are abducted or bred for [[w:Human_sacrifice|human sacrifices]], [[w:Child_pornography|pornography]], and [[w:Prostitution|prostitution]].

The key elements of the narrative are:
*Satanic ritual abuse is horrific and widespread.
*Unknown to us many of our children are being subjected to the horrors of satanic ritual abuse.
*The abuse is so horrible that the children [[w:Repressed_memory|repress their memories]] of the abuse and are unable to disclose their experiences.
*A newly developed interviewing technique, called [[w:recovered-memory_therapy|recovered-memory therapy]], can elicit accurate memories and testimony from the children.
*Using this technique, many children are beginning to reveal and describe the abuse they have suffered.
*This must be urgently investigated, and the abuse stopped at all costs.
*A global Satanic cult may be responsible.
*Missing memories among the victims and absence of evidence was cited as evidence of the power and effectiveness of the cult in furthering their agenda.

Initial interest arose via the publicity campaign for Pazder's 1980 book [[w:Michelle_Remembers|''Michelle Remembers'']], and it was sustained and popularized throughout the decade by coverage of the [[w:McMartin_preschool_trial|McMartin preschool trial]] and the contemporaneous [[w:Day-care_sex-abuse_hysteria|day-care sex-abuse hysteria]]. Testimonials, symptom lists, rumors, and techniques to investigate or uncover memories of SRA were disseminated through professional, popular, and religious conferences, as well as through [[w:Talk_show|talk shows]], sustaining and further spreading the moral panic throughout the United States and beyond. In some cases, allegations resulted in criminal trials with varying results; after seven years in court, the McMartin trial resulted in no convictions for any of the accused, while other cases resulted in lengthy sentences, some of which were later reversed. Scholarly interest in the topic slowly built, eventually resulting in the conclusion that the phenomenon was a moral panic, which, as one researcher put it in 2017, &quot;involved hundreds of accusations that devil-worshipping pedophiles were operating America's white middle-class suburban daycare centers.&quot; 

Of the more than 12,000 documented accusations nationwide, investigating police were not able to substantiate any allegations of organized cult abuse.

Today the [[w:Radical_right_(United_States)|far-right]] conspiracy theory movement known as [[w:QAnon|QAnon]], has adopted many of the tropes of Satanic Ritual Abuse and Satanic Panic. Instead of daycare centers being the center of abuse, however, liberal [[w:Hollywood|Hollywood]] actors, [[w:Democratic_Party_(United_States)|Democratic]] politicians, and high-ranking government officials are portrayed as a child-abusing cabal of Satanists.
=== Assignment ===
'''Part 1:'''

#Identify a powerful false narrative to study for this assignment. Choose one from this list of [[Finding Common Ground/Powerful False Narratives|powerful false narratives]], or from any other source.
#Identify the elements that make this narrative compelling, convincing, memorable, and likely to spread and be shared with others.
#Identify the falsehoods in the narrative.
#Identify any [[Embracing Ambiguity|ambiguities]] that are prematurely resolved. 
#Identify the various calls to action inspired by the narrative.
#How is this narrative harmful, if at all? 
#Who gains and who loses as this narrative spreads?

'''Part 2:'''

#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of narrative, language, perception, or reality.
#If the narrative is driving the conversation, go through the steps above to analyze the narrative elements.
#Practice dialogue rather than debate or argumentation. Be [[w:Candor|candid]]. [[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course. 
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[Evaluating Evidence|research them]].
##[[Seeking True Beliefs|Seek true beliefs]].
##[[Embracing Ambiguity|Embrace ambiguity]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Ideology ==
[[File:Ideology Occludes Reality.jpg|thumb|Ideology Occludes Reality]] 
Do you think for yourself and choose your own beliefs? If you are like most people, you probably find it easier to adopt some pre-packaged set of beliefs that seem attractive. We can broadly characterize an [[w:Ideology|ideology]] as some agenda-driven set of beliefs.&lt;ref&gt;Many definitions of ideology are proposed. See, for example: [https://journals.sagepub.com/doi/10.1177/106591299705000412 Ideology: A definitional Analysis], John Gerring, December 1, 1997, Political Research Quarterly. &lt;/ref&gt;  Because ideologies are constructed to serve some agenda, an ideology is unlikely to accurately represent reality. 

An [[w:ideology|ideology]] is a set of beliefs intended to describe how the world works, or how some believe it should work. An ideology is a particular way of looking at the world, often codified into a [[w:doctrine|doctrine]]. Often our religious, political, and economic beliefs are drawn from an ideology. You may also follow particular lifestyle choices such as [[w:veganism|veganism]], or [[w:Environmentally_friendly|environmentalism]] based on a particular ideology. Ideologies substitute socially constructed models for brute facts. Many ideological models do not correspond well to reality. “Essentially, all models are wrong,” [[w:George_E._P._Box|George Box]] noted, “but some are useful.” Beware of substituting an ideology for a careful examination of reality.

As illustrated in the revised diagram shown here, ideologies impose a model that prohibits direct access to reality, and displaces any alternative narratives, interpretations, representations, or perceptions of reality. The ideology establishes all you need to know. It acts as a convenient substitute for reality. 

Immersion and commitment to an ideology can become a firmly held part of your identity. If you say “I am a Conservative” rather than “I often agree conservative political ideas” you are declaring the ideology as a part of your identity.&lt;ref&gt;{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant|date=February 2, 2021 |title=Think Again: The Power of Knowing What You Don't Know  |publisher=Viking |pages=320 |isbn=978-1984878106}}&lt;/ref&gt; This can make it harder to abandon. Although you choose your beliefs, [[True_Self#I_am|you are not your beliefs]]. 

Remaining bound by an ideological doctrine is a form of mental bondage. It is wise to break free from that bondage. Adopting a scout mindset—described below—can help us break free from ideologies that are holding our minds captive.

=== The Scout Mindset ===
[[File:U.S. Marine and Japan Ground Self-Defense Scout Snipers 170310-M-PQ336-010.jpg|thumb|Scouts seek to see the world as it is, not as they wish it was.]]
Author [[w:Julia_Galef|Julia Galef]] describes the ''scout mindset'' as “The motivation to see things as they are, not as you wish they were.”&lt;ref&gt; {{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}} @9 of 541&lt;/ref&gt;  In contrast to the ''scout mindset'', the ''soldier mindset'' is a motivation to attack differing points of view or defend a position in each argument we encounter. 
 
Contrasted with the soldier mindset, the scout mindset values:
*'''Seeing things as they''' are over defending territory or taking territory;
*'''Asking is it true?''' over arguing to defend a pre-determined position;
*'''Observations''' over interpretations;
*'''Investigation''' over argumentation;
*'''Gaining insight''' over winning this argument;
*'''Wonder''' over attacking;
*'''Learning''' over advancing a position;
*'''Observing''' over taking ground;
*'''Understanding the issue''' over shooting down arguments;
*'''Exploring possibilities''' over reinforcing a position;
*'''[[Seeking True Beliefs|Seeking true beliefs]]''' over securing long held beliefs;
*'''Seeking reality''' over defending an ideology;
*'''[[Evaluating Evidence|Objective evidence]]''' over motivated reasoning;
*'''Representative evidence''' over narratives, specious interpretations and representations;
*'''Reason''' over rhetoric;
*'''Exploration''' over staying on the ideological course;
*'''Dialogue''' over reciting [[w:Dogma|dogma]];
*'''Listening''' over reiterating and continuing to advocate;
*'''[[Virtues/Humility|Humility]]''' over arrogance;
*'''Curiosity''' over certainty or fear;
*'''[[Intellectual honesty]]''' over [[w:Prevarication|prevarication]], and
*'''Working with collaborators''' over fighting with opponents.

[[Coping with Ego|Ego]] encourages the soldier. [[Deductive_Logic/Clear_Thinking_curriculum|Reason]] encourages the scout.

The [[w:Stanford_Encyclopedia_of_Philosophy|Stanford Encyclopedia of Philosophy]] entry on Law and Ideology tells us, “Ideologies are ideas whose purpose is not epistemic, but political. Thus, an ideology exists to confirm a certain political viewpoint, serve the interests of certain people, or to perform a functional role in relation to social, economic, political, and legal institutions.” &lt;ref&gt;Sypnowich, Christine, &quot;[https://plato.stanford.edu/archives/sum2019/entries/law-ideology Law and Ideology]&quot;, The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.) &lt;/ref&gt;

Because ideologies exist to advance an agenda rather than to model reality, they require a soldier mindset to sustain the gap between the model described by the ideology and an accurate model based on reality. Ideologies are vulnerable to exploration by a scout mindset. Adopting a scout mindset is key to escaping ideologies.

=== Example Ideologies ===
We rely on many ideologies to simplify our thinking. These include: 
*[[w:Loyalty|Loyalty]], including loyalty to an individual person, a group, team, brand, causes, regions, nation, or any of the belief systems listed below. 
*[[w:List_of_mythologies|Myths and legends]],
*[[w:Folklore|Folklore]], [[w:Tradition|traditions]], [[w:Superstition|superstitions]], and [[w:Taboo|taboos]],
*[[w:Caste|Caste systems]], including [[w:Racism|racism]], [[w:Sexism|sexism]], other forms of [[w:Ascribed_status|ascribed status]], [[w:Social_class|social classes]], and other ranking constructs, 
*[[w:Stereotype|Stereotypes]], 
*[[w:Economic_ideology|Economic ideologies]], including [[w:Neoliberalism|neoliberalism]], [[w:Monetarism|monetarism]], [[w:Mercantilism|mercantilism]], [[w:Mixed_economy|mixed economy]], [[w:Social_Darwinism|social Darwinism]], [[w:Communism|communism]], [[w:Laissez-faire|laissez-faire economics]], and [[w:Free_trade|free trade]]. There are also current theories of [[w:Safe_trade|safe trade]] and [[w:Fair_trade|fair trade]] that can be understood as ideologies.
*[[w:List_of_political_ideologies|Political ideologies]], including anarchism, authoritarianism, communism, conservatism, democracy, environmentalism, fascism, separatist movements, liberalism, libertarianism, nationalism, populism, social democracy, socialism, and others.
*[[w:List_of_creation_myths|Creation myths]],
*[[w:List_of_religions_and_spiritual_traditions|Religions and spiritual traditions]], 
*[[w:Quackery|Quackery]],
*[[w:List_of_topics_characterized_as_pseudoscience|Pseudoscientific beliefs]],
*[[w:Paranormal|Paranormal beliefs]],
*Commitment to [[w:List_of_conspiracy_theories|conspiracy theories]], and
*[[w:List_of_new_religious_movements|New religious movements]].
=== Assignment ===
'''Part 1:'''

#Identify each of the ideologies you identify with, belong to, or agree with. Use the list above as a guide or use any other method to identify ideologies that influence you.
#For each of the ideologies identified in step 1:
##Decide if the model it presents is an accurate representation of reality in all its scope and complexity.
##Identify any [[Embracing Ambiguity|ambiguities]] that are prematurely resolved. 
##Decide if the ideology is helping you understand reality, or is occluding, limiting, biasing, or censoring your understanding of reality.
##If you decide a particular ideology is unhelpful, take steps to abandon that ideology. 

Welcome those who are [[Seeking True Beliefs|seeking truth]]. Abandon those who are [[w:Certainty|certain]] they have found [[w:Dogma|Truth]].

'''Part 2:'''
#Study the module on [[Knowing_How_You_Know/Examining_Ideologies|Examining Ideologies]] within the [[Knowing How You Know|Knowing how you know]] course.
#Complete the [[Knowing_How_You_Know/Examining_Ideologies#Assignment|assignments]] in that module.
#Read the essay [[/Every Ism Creates a Schism/|&quot;Every Ism Creates a Schism&quot;: An Exploration]].
#Think beyond the doctrine.

== Toward Ought ==
[[File:Compass rose browns 00.png|thumb|right| 250px|A deep understanding of [[Moral_Reasoning#A_Basis_for_Moral_Reasoning |''impartiality'']] can guide toward what we ought to do. ]]
So far in this course, the common ground we have considered is our shared reality. This is our collective understanding of ''what is'' in the world. Can we find a corresponding common ground regarding what we ''ought to do''?

Philosopher [[w:David_Hume|David Hume]] famously observed that knowing only ''what is'', we cannot determine what we [[w:Is–ought problem|''ought to do'']]. However, by making the reasonable [[Moral_Reasoning#A_Basis_for_Moral_Reasoning |assumption of ''impartiality'']], we can begin to identify what it is we ought to do. 

We ought to [[Assessing_Human_Rights/Beyond_Olympic_Gold|advance human rights worldwide]], adopt [[Level_5_Research_Center#Values|pro-social values]], and establish a well-founded basis for [[Moral Reasoning|moral reasoning]].

=== Assignment ===
#Complete the Wikiversity course on [[Assessing Human Rights]].
#[[Assessing_Human_Rights/Beyond_Olympic_Gold|Advance human rights, worldwide]]. 
#Study this [[Level_5_Research_Center#Values|list of pro-social values]].
#Adopt pro-social values and [[Level_5_Research_Center/Choosing_Level_5_Living|choose level 5 living]]. 
#Complete the Wikiversity course on [[Moral Reasoning]].
#Develop your own well-founded basis for [[Moral Reasoning|moral reasoning]].

== Simple but not easy ==
The central idea in this course—[[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]]—is simply stated and supported by overwhelming evidence. However, it may not be easy for you, or others you encounter or care about to change deeply held beliefs that differ from objective reality. Here are some steps you can take to align your beliefs with reality and work to find common ground with others.
#Begin by aligning your own beliefs with reality. Complete the course on [[Seeking True Beliefs]]. [[Exploring_Worldviews/Aligning_worldviews|Align your own worldview]] with reality. [[Deductive_Logic/Clear_Thinking_curriculum|Think clearly]]. [[Knowing How You Know|Know how you know]]. Become skillful at [[Evaluating Evidence|evaluating evidence]]. Stay curious. [[Embracing Ambiguity|Embrace ambiguity]].
#Be careful to [[Facing_Facts#Degrees_of_Consensus|distinguish among matters of fact]], matters of controversy, and matters of opinion. Do not argue matters of fact, research them. Do not argue matters opinion, enjoy them. Reason carefully and listen closely when discussing matters of controversy. Although [[Virtues/Tolerance|tolerance]] is essential in matters of opinion, it has no place in [[Facing Facts|matters of fact]].
#Spend the required effort to prepare to find common ground with others. 
##Complete the course on [[Practicing Dialogue|practicing dialogue]]. Practice dialogue.
##Ensure all participants have adopted a [[Socratic_Methods#Essential_Socratic_Temperament|Socratic temperament]].
##Expect [[intellectual honesty]].
###Complete the course on [[intellectual honesty]].
###[[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
###Discuss the importance of [[intellectual honesty]]—accurately describing your true beliefs—with each participant. Gain agreement to be intellectually honest, and to expect intellectual honesty from all participants. Do not indulge [[w:Charlatan|charlatans]]. Do not [[w:Internet_troll|feed the trolls]]. 
#Finding common ground often challenges deeply held beliefs. This is uncomfortable; people often react with fear and may seek to withdraw from the dialogue or divert the conversation.  Notice the shift from [[Finding_Common_Ground#The_Scout_Mindset|scout mindset]] to soldier mindset. Notice when fear begins to displace curiosity. Encourage the fearful person to allow curiosity to displace fear. Return to a scout mindset.
#Real good is our common ground. People who are [[Living Wisely/Seeking Real Good|seeking real good]] will find common ground with others who are also seeking real good. When [[Transcending Conflict|conflict arises]], it is likely because someone is not seeking real good. 
#The course on [[Street Epistemology]] provides specific techniques for exploring the basis for beliefs. Use applicable Street Epistemology techniques when seeking common ground.

== Summary and Conclusions ==
Reality exists. Reality is vast, complex, and dynamic. Humans have only investigated a small portion of the universe, and our investigation is incomplete. We all live together on this one planet we call Earth and [[Knowing_How_You_Know/One_World|we all live in the same universe]].

Because we all live in the same universe, our reliable understanding of that universe must eventually converge toward one coherent description. Because reality exists, we can [[Evaluating_Evidence|examine reality]], and we can [[Exploring_Worldviews/Aligning_worldviews|align our worldview]] with reality. Building on these observations, it follows that [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]], even though any number of worldviews is possible.

[[w:Perception|Perception]] transforms sensory information originating with some material object, known as the ''target'' or the ''[[w:Distal|distal]] stimulus'' into mental representations known as [[w:Perception#Process_and_terminology|percepts]], which do not exist elsewhere in the world. 

[[w:Perception|Perception]]—the process of extracting information from energy that impinges on our [[w:Sense|sensory organs]]—is not straightforward. There is more to perception than meets the eye.

Our most direct encounters with reality are through direct sensory observations and awareness. However, our perceptions are not exact replicas of the distal objects being perceived. Omissions, distortions, and additions take place. Fortunately, we can augment our direct perceptions to obtain a more accurate understanding of our world. Furthermore, [[Facing_Facts/Perceptions_are_Personal|perceptions are personal]], and it is often a [[w:Problem_of_induction|mistake to generalize]] our personal perceptions beyond our own experiences.

We represent our thoughts using various symbols, including words, gestures, facial expressions, images, and analogies. The mapping from various representations to reality is imprecise. Language and other symbols can be constraining, ambiguous, refer to social constructs, prejudiced, based on artificial boundaries, or focus on convenient representation rather than reality. The map is not the territory, although we often confuse our representations for realty.

People [[w:Interpretation_(philosophy)|interpret]] observations to resolve ambiguity, increase certainty, and to account for observations within some familiar model or analogy. The role of interpretation is most evident when the items being interpreted are most [[w:Ambiguity|ambiguous]]. 

[[w:Tasseography|Tasseography]]—reading tea leaves—reveals the essence of interpretation. Because the tea leaf formations are entirely arbitrary the interpreter can say almost anything. The interpretation reflects judgements, evaluations, biases, concepts, models, analogies, predictions, optimism, pessimism, certainty, risk, doubt, hope, fears, storytelling skills, and desires of the interpreter, entirely independent of the tea leaf formation.

The [[w:Rabbit–duck illusion|rabbit-duck illusion]] allows for two different, yet equally correct interpretations. Life is often more complicated than ducks and rabbits, and we can reasonably differ in interpreting many words.

Interpretations that prematurely eliminate [[w:Doubt|doubt]] and resolve [[w:Ambiguity|ambiguity]] into a comfortable feeling of [[w:Certainty|certainty]] have led to several tragedies. Become [[Embracing Ambiguity|comfortable with ambiguity]].

Humans enjoy telling and retelling [[w:Narrative|stories]]. This may be the defining characteristic of the human species. We are exposed to many powerful false narratives. To find common ground we must dismiss the falsehoods in narratives.

An [[w:ideology|ideology]] is a set of beliefs intended to describe how the world works, or how some believe it should work. An ideology is a particular way of looking at the world, often codified into a [[w:doctrine|doctrine]]. Often our religious, political, and economic beliefs are drawn from an ideology.

Ideologies substitute socially constructed models for brute facts. Many ideological models do not correspond well to reality. “Essentially, all models are wrong,” [[w:George_E._P._Box|George Box]] noted, “but some are useful.” Beware of substituting an ideology for a careful examination of reality.

Remaining bound by an ideological doctrine is a form of mental ''bondage''. It is wise to break free from that bondage. Adopting a ''scout mindset'' can help us break free from ideologies that are holding our minds captive.

It is likely your beliefs are influenced by [[Knowing How You Know/Examining Ideologies|inaccurate ideologies]]. Identify these and abandon them.

This is not about compromise. This is about gaining an accurate understanding of the world we live in.

[[Facing Facts/Reality is the Ultimate Reference Standard|Reality is our reference standard]]. 

Embrace reality. Seek true beliefs. Dismiss misleading perceptions, representations, interpretations, narrations, and ideologies. Align concepts with reality.

[[Assessing_Human_Rights/Beyond_Olympic_Gold|Advance human rights worldwide]], adopt [[Level_5_Research_Center#Values|pro-social values]], and establish a well-founded basis for [[Moral Reasoning|moral reasoning]].

[[Transcending_Conflict|Transcend conflict]].

[[Living_Wisely/Seeking_Real_Good|Seek real good]].

It is difficult to change deeply held beliefs, however if ''you'' can do this, ''they'' can do this.

We can [[Facing_Facts/Reality_is_our_common_ground|find common ground]].

== Recommended Reading ==
*{{Cite book|title=Truth: what it is, how to find it, and why it still matters|publisher=Johns Hopkins University Press|date=2026|location=Baltimore|isbn=978-1-4214-5372-9|first=Michael|last=Shermer}}
*{{cite book |last=Van der Stigchel |first=Stefan |date=March 12, 2019 |title=How Attention Works: Finding Your Way in a World Full of Distraction |publisher=The MIT Press |pages=152 |isbn=978-0262039260 }}
*{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }} 
*{{cite book |last1=Gause  |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is  |publisher=Dorset House Publishing Company |pages=176 |isbn=978-0932633163 |author-link=w:Gerald_Weinberg}} 
*{{cite book |last=Weinberg |first=Gerald M. |author-link=w:Gerald_Weinberg |date=September 1, 1989 |title=Exploring Requirements: Quality Before Design  |publisher=Dorset House Publishing Company |pages= 320 |isbn=978-0932633132}}
*{{cite book |last=Holmes |first=Jamie |author-link=w:Jamie_Holmes_(author) |date=October 11, 2016 |title=Nonsense: The Power of Not Knowing Paperback  |publisher=Crown |pages=336 |isbn=978-0385348393}}
*{{cite book |last=Ariely |first=Dan |author-link=w:Dan_Ariely |date=September 17, 2024 |title=Misbelief: What Makes Rational People Believe Irrational Things |publisher=Harper Perennial  |pages=320 |isbn=978-0063280434}}
*{{cite book |last=Burton M.D. |first=Robert A.  |date=Mar 17, 2009 |title=On Being Certain: Believing You Are Right Even When You're Not |publisher=St. Martin's Griffin |pages272 |isbn=978-0312541521}}
*{{cite book |last=Duke |first=Annie |author-link=w:Annie_Duke |date= |title=Thinking in Bets: Making Smarter Decisions When You Don't Have All the Facts |publisher=Portfolio |pages= 288 |isbn=978-0735216372}}
*{{cite book |last=Freinacht  |first=Hanzi  |date=March 10, 2017 |title=The Listening Society: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=414 |isbn=978-8799973903}}
*{{cite book |last=Freinacht  |first=Hanzi  |date=May 29, 2019 |title=Nordic Ideology: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=495 |isbn=978-8799973927}}
*{{cite book |last1=Gilovich |first1=Thomas |last2=Ross |first2=Lee |date=December 1, 2015 |title=The Wisest One in the Room: How You Can Benefit from Social Psychology's Most Powerful Insights|publisher=Free Press|pages=320|isbn=978-1451677546}} 
*{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}}
*{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}
*{{cite book |last1=Hofstadter |first1=Douglas R  |last2=Sander |first2=Emmanuel |author-link=w:Douglas_Hofstadter |date=April 23, 2013 |title=Surfaces and Essences: Analogy as the Fuel and Fire of Thinking  |publisher=Basic Books |pages=592 |isbn=978-0465018475}}
*{{cite book |last= Weinberg |first=Gabriel  |date=June 18, 2019 |title=Super Thinking: The Big Book of Mental Models |publisher=Portfolio |pages=352  |isbn=978-0525533580}}
*{{cite book |title=The Art of Possibility: Transforming Professional and Personal Life |last1=Stone Zander |first1=Rosamund |last2=Zander|first2=Benjamin   |year=224 |publisher=Penguin |isbn=978-0142001103  |pages=224}}
*{{cite book |last1=Lakoff |first1=George |last2=Johnson |first2=Mark|date=April 15, 2003 |title=Metaphors We Live By |publisher= |pages=242 |isbn=978-0226468013 |author-link=w:George Lakoff }}
*{{cite book |last1=Heath |first1=Chip |last2=Heath |first2=Dan  |author-link=w:Chip_Heath|date= |title=Made to Stick: Why Some Ideas Survive and Others Die   |publisher=Random House|pages= 291 |isbn=978-1400064281}} 
*{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant|date=February 2, 2021 |title=Think Again: The Power of Knowing What You Don't Know  |publisher=Viking |pages=320 |isbn=978-1984878106}} 
*{{cite book |last1=Campbell |first1=Joseph | last2=Moyers |first2=Bill |date=June 1, 1991 |title=The Power of Myth  |publisher=Anchor |pages=293 |isbn=978-0385418867 |author-link=w:Joseph_Campbell }}
*{{cite book |last=Mackay |first=Charles |date=November 1, 2016 |title=[[w:Extraordinary Popular Delusions and The Madness of Crowds|Extraordinary Popular Delusions and The Madness of Crowds]] |publisher=CreateSpace Independent Publishing Platform |pages=386 |isbn=978-1539849582 |author-link=w:Charles_Mackay_(author) }}
*{{cite book |last=Orwell |first=George  |date=March 1, 2017 |title=[[w:Animal Farm|Animal Farm]] |publisher=Fingerprint! Publishing |pages=152 |isbn=978-9386538284 |author-link=w:George_Orwell }}
*{{cite book |last=Orwell  |first=George |date=January 15, 2013 |title=[[w:Politics and the English Language|Politics and the English Language]] |publisher=Penguin Classic |pages=48 |isbn=978-0141393063 |author-link=w:George_Orwell }}
*{{cite book |last=Hecht |first=Jennifer Michael |author-link=w:Jennifer_Michael_Hecht |date=September 7, 2004 |title=Doubt: A History: The Great Doubters and Their Legacy of Innovation from Socrates and Jesus to Thomas Jefferson and Emily Dickinson |publisher=HarperOne |pages=576 |isbn=978-0060097950}}
*{{cite book |last=Carroll |first=Sean M |author-link=w:Sean_M._Carroll |date=May 4, 2017 |title=The Big Picture: On the Origins of Life, Meaning, and the Universe Itself  |publisher=Oneworld Publications |pages=480 |isbn=978-1786071033}}
*{{cite book |last=Pinker |first=Steven |author-link=w:Steven_Pinker |date=February 13, 2018 |title=Enlightenment Now: The Case for Reason, Science, Humanism, and Progress |publisher=Viking |pages=576  |isbn=978-0525427575}}
*{{cite book |last=Wilczek |first=Frank |author-link=w:Frank_Wilczek |date=January 12, 2021 |title=Fundamentals: Ten Keys to Reality  |publisher=Penguin Press |pages=272 |isbn=978-0735223790}}
*{{cite book |last=Gray |first=Dave |author-link= |date=September 14, 2016 |title=Liminal Thinking: Create the Change You Want by Changing the Way You Think  |publisher=Two Waves Books |pages=184 |isbn=978-1933820460}}
*{{cite book |last=Schulz |first=Kathryn |author-link=w:Kathryn_Schulz |date=June 8, 2010 |title=Being Wrong: Adventures in the Margin of Error |publisher=Ecco |pages=416 |isbn=0061176044}}
*{{cite book |last=Temple |first=David J.  |date=April 2, 2024 |title=First Principles and First Values: Forty-Two Propositions on Cosmoerotic Humanism, the Meta-Crisis, and the World to Come  |publisher=World Philosophy &amp; Religion Press |pages=296 |isbn=979-8989588909}}

I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research.

*{{cite book |last=Coleman |first=Peter T. |author-link=w:Peter_T._Coleman_(academic) |date=June 1, 2021 |title=The Way Out: How to Overcome Toxic Polarization  |publisher=Columbia University Press |pages=296 |isbn=978-0231197403}}

== References ==
&lt;references/&gt;

{{CourseCat}} 
[[Category:Life skills]]
[[Category:Applied Wisdom]]
[[Category:Philosophy]]
[[Category:Clear Thinking]]
[[Category:Courses]]
[[Category:Community]]
[[Category:Reality]]
[[Category:Reformation Workshop]]
{{Clear Thinking}}</text>
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      <text bytes="101601" sha1="1id13b4esucuxyg7icocleuhfchj61y" xml:space="preserve">— Aligning concepts with reality.
[[File:Layers of Abstraction.jpg|thumb|We engage reality at various layers of abstraction.&lt;ref&gt; This diagram is used as the primary organizing structure for the course. Each ring in the diagram corresponds to a section of the course. Notice that each boundary is blurred. This blurring acknowledges that the various layers interact and no sharp boundary between layers exists. &lt;/ref&gt;]] 
{{TOC right | limit|limit=2}} 
== Introduction ==
Although we all live in the same world and share a single reality, we often seem to be worlds apart when discussing important issues. What is going on? How can we find common ground?&lt;ref&gt;According to [[w:Aumann's_agreement_theorem|Aumann's agreement theorem]], we will be able to find common ground.&lt;/ref&gt;
== Objectives ==
{{100%done}}{{By|lbeaumont}}
The objectives of this course are to:
*Understand the nature of reality.
*Identify the many layers of abstraction through which we encounter reality.
*Navigate through these layers of abstraction.
*Diagnose reasons for conflict during discussions.
*Find common ground.

This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]], the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking curriculum]], and the [[Coming Together]] curriculum.

Use this [[/Daily Practice: Finding Common Ground/|daily practice checklist]] to make finding common ground a habit. 

[https://www.academia.edu/106815969/Finding_Common_Ground Slides based on this course] are available, along with a [https://youtu.be/pQMEDRVwI5U video presentation] of those slides. 

If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont|click here to send me an email]] or leave a comment or question on the [[Talk: Finding Common Ground|discussion page]].

[[File:Finding Common Ground Audio Dialogue.wav|thumb|Finding Common Ground Audio Dialogue]]

== Importance ==
Finding common ground is an important skill because it is useful for resolving the [[w:Social_polarization|social]], [[w:Culture_war|cultural]], [[w:Political_polarization|political]], and [[w:Economic_inequality|economic]] polarization that is prevalent. [[w:Religious_violence|Religious conflicts]], [[w:War|international conflicts]], [[w:Nationalism|nationalism]], [[w:Class_conflict|class struggles]], [[w:Ethnic_conflict|racial tensions]], [[w:Culture_war|culture wars]], ideological conflicts, [[w:Gender_inequality|gender inequality]], [[w:Political_polarization|political polarization]], [[w:Book_censorship|book banning]], [[w:Hate_crime|hate crimes]], [[w:Climate_change_denial|climate change denial]], and [[w:Conspiracy_theory|conspiracy theories]] are a few examples of the conflicts that are dividing people throughout the world.

When we can recognize that [[Facing Facts/Reality is our common ground|reality is our common ground]], we can restore the [[Virtues/Civility|civility]] that is essential to peaceful coexistence. 

== In a Nutshell ==
Here is a brief introduction to the key concepts presented in this course. [[Finding_Common_Ground#Introduction|The diagram]] can help you organize, recall, share, and apply these ideas.
#[[Finding_Common_Ground#Reality|Reality]] exists and we all share in a single objective reality.&lt;ref&gt;{{Cite web|url=https://substack.com/inbox/post/174781036|title=A Minimum Viable Metaphysics|last=Substack|website=substack.com|language=en|access-date=2025-09-30}}&lt;/ref&gt;
#There are several reasons why this fact is difficult for us to grasp and hold onto.
##[[Finding_Common_Ground#Reality|Reality]] is vast, complex, and dynamic. Each of us directly encounters only a tiny slice of reality. We each experience only a glimpse of our vast universe.
##Our direct contact with reality is through our [[Finding_Common_Ground#Perception|perceptions]], which introduce omissions, distortions, and additions.
##Although we use words and other symbols to [[Finding_Common_Ground#Representation|represent]] reality as we perceive it, these symbols are limited, and they only provide approximate representations of our perceptions.
##Because much of what we encounter is [[Embracing Ambiguity|ambiguous]], it invites us to [[Finding_Common_Ground#Interpretation|interpret]] the information to resolve the ambiguity and provide us the comfort of certainty. Many cognitive biases influence our interpretations.
##We love telling and retelling [[Finding_Common_Ground#Narration|stories]]. We easily substitute alluring stories for the complexities and difficulties of reality.
##[[Finding_Common_Ground#Ideology|Ideologies]] substitute a simplified belief system for the complexities of reality. We are easily attracted to these easy to use explanations.
#We can see beyond these illusions and better comprehend reality.
#Reality is our common ground. We can each find that common ground by advancing toward the center of [[Finding_Common_Ground#Introduction|the diagram]] shown above.
#Although these ideas are simply stated, they are difficult to fully grasp and put into practice. Please complete the remainder of this course and use these insights every day.
#We can find common ground.

== Reality ==
For many practical reasons, this course begins with the assumption that [[w:Reality|reality]] exists. Everyday experience provides [[Evaluating Evidence|evidence]] that reality exists. Every time you decide to open the door before passing through the doorway, you are betting that reality exists. If you have lost your keys, then opening the door, or starting your car can become a real problem. If you have difficulty levitating, leaping tall buildings in a single bound, seeing through brick walls, teleporting, or time travelling, perhaps it is because you are encountering constraints imposed by reality.

Despite empirical evidence, people often [[Does objective reality exist?|argue against the existence of reality]]. In these arguments people may cite the [[w:Allegory_of_the_cave|allegory of the cave]], the [[w:Brain_in_a_vat|brain in a vat]], the [[w:Simulation_hypothesis|simulation hypothesis]], the [[w:The_Matrix_(franchise)|Matrix movies]], claims that [[Facing_Facts/Perceptions_are_Personal|perception is reality]], and [[w:Postmodernism|postmodern theories]].

While these are fascinating thought experiments that do deserve some serious philosophical investigation, they don’t provide much help in getting through our daily lives. I bet that reality exists.&lt;ref&gt; Although I am very confident that reality exists, no one can be [[w:Certainty|certain]]. Much of this course demonstrates the value of embracing and exploring [[w:Doubt|doubt]] while deferring certainty. Also, it is often [[Wisdom|wise]] to think in bets. See: {{cite book |last=Duke |first=Annie |author-link=w:Annie_Duke |date= |title=Thinking in Bets: Making Smarter Decisions When You Don't Have All the Facts |publisher=Portfolio |pages= 288 |isbn=978-0735216372}} &lt;/ref&gt;

For the remainder of the course, we proceed with the assumption that reality exists.

Reality is vast, complex, and dynamic. Humans have only investigated a small portion of the universe, and our investigation is incomplete. Our awareness, observations, and perceptions of reality are neither complete nor accurate representations of reality. We do not observe the many [[w:Cosmic_ray|cosmic rays]] passing through us each second, the atoms that make up the materials we encounter, billions of galaxies beyond the limits of our vision, the ultrasonic chirps used by bats to navigate, viruses, DNA, antibodies, [[w:Greenhouse_gas |greenhouse gasses]], [[w:Particulates|particulate contaminants]], and much more of reality as it is.

This course makes the further assumption that [[Knowing_How_You_Know/One_World|we all live in the same universe]]&lt;ref&gt;This assumption does not conflict with [[w:Many-worlds_interpretation|many-worlds interpretations]] of quantum mechanics. &lt;/ref&gt;. All life forms discovered so far live together on our single planet, circling our sun, in our humble place in the universe. The universe is vast, yet it is all one world, and we all live together on this one planet we call Earth. All that we know of and all that we have ever experienced follow the same laws of physics. Remarkably, the entire world as we know it has emerged from a few [[w:Standard_Model|fundamental building blocks]].

Because we all live in the same universe, our reliable understanding of that universe must eventually converge toward one coherent description. Each phenomenon we observe must fit into a single coherent and integrated description of our universe. Either the description must evolve to accommodate each new observation, or our understanding of that observation must be interpreted consistently with that unified representation.

Building on these assumptions, it follows that [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]], even though any number of [[Exploring Worldviews|worldviews]] is possible. It is likely that each of us hold a worldview that is somewhat different from others. Of all the possible worldviews, one worldview is especially important. That is the worldview that corresponds to reality as closely as our best current understanding of reality allows. Because reality exists, we can [[Evaluating_Evidence|examine reality]], and we can [[Exploring_Worldviews/Aligning_worldviews|align our worldview]] with reality. Because we all live in one world, reality is our common ground.
=== Assignment ===
#Complete the Wikiversity course on [[Facing Facts]].
#Read the essay [[Facing_Facts/Perceptions_are_Personal|Perceptions are personal]].
#Investigate further any of the arguments that appeal to you, listed above, that reality does not exist.
#If you can accept the assumption that reality exists, please proceed with the remainder of this assignment.
#Estimate what fraction of reality you are familiar with.
##Read the essay, [[Virtues/Humility/Being_99.9%_Ignorant|Being 99.9% Ignorant]].
##How many countries are there in the world? How many have you visited?
##Study modern research on the [[w:Galaxy#Modern_research|number of galaxies in the universe]]. How many have you visited?
##Scan this list of [[w:Lists_of_unsolved_problems|lists of unsolved problems]]. What fraction of reality has been investigated by humans?
##Can you accept the premise that reality exists far beyond our perceptions, investigations, and conceptions of it?
#Read the essay [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]].
#Read the essay [[Knowing_How_You_Know/One_World|One World]].
#Read the essay [[Facing_Facts/Reality_is_our_common_ground|Reality is our common ground]].
#Read the essay [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]].
#Optionally read the essay [[Beyond_Theism/What_there_is|What there is]]. Consider writing your own such essay.  
#Can you accept the premise that reality exists, we live in one world, we share one reality, and that reality  is our common ground? If so, please proceed with the remainder of this course.

== Perception ==
[[w:Perception|Perception]] transforms sensory information originating with some material object, known as the ''target'' or the ''[[w:Distal|distal]] stimulus'' into mental representations known as [[w:Perception#Process_and_terminology|percepts]], which do not exist elsewhere in the world. 

[[w:Perception|Perception]]—the process of extracting information from energy that impinges on our [[w:Sense|sensory organs]]—is not straightforward. There is more to perception than meets the eye.&lt;ref&gt;{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }}Page 13 of 162.&lt;/ref&gt;

Indirect theories of perception describe it as a constructive process that involves inference, learning, and experience.&lt;ref&gt;{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }}Page 13 of 162.&lt;/ref&gt; 

While it may seem reasonable that an ideal perception system would duplicate the observed target object exactly in the mental percept, this is not what happens. One theory hypothesizes that the purpose of perception is to allow organisms to locate and use [[w:Affordance|affordances]]—what the environment offers the individual—for practical benefit.

Our most direct encounters with reality are through direct sensory observations and awareness. However, our perceptions are not exact replicas of the distal objects being perceived. Omissions, distortions, and additions take place. Fortunately, we can augment our direct perceptions to obtain a more accurate understanding of our world.

=== Omissions ===
Our direct perception accounts for only a tiny fraction of reality.

We can only directly perceive stimuli that our senses encounter, and these senses experience only a small fraction of reality. We do not see what is behind us, the places we do not visit, or stimuli outside the range of our sensory systems. Attention is selective, so we are likely to perceive only what our attention is drawn to, remaining unaware of the rest. When our attention is captured by moving objects, shiny objects, or unexpected occurrences, we may be distracted from sensing other information in our environment.
[[File:Directing_Attention.jpg|300px|thumb|Our perceptions focus on what we direct our attention toward.]]
Although reality is our common ground, each of us has unique experiences of its vastness. Refer to the diagram on the right. Reality is vast, complex, and dynamic, and the range of experiences any of us has is a tiny fraction of reality as it is. Furthermore, within our range of experience, our attention is limited. Our attention concentrates awareness on some phenomena to the exclusion of other stimuli.&lt;ref&gt;The idea of representing attention as a vertical line sampling reality is presented in {{cite book|title=Liminal Thinking: Create the Change You Want by Changing the Way You Think  |last=Gray|first=Dave|date=September 14, 2016|publisher=Two Waves Books|isbn=978-1933820460|pages=184|author-link=}}&lt;/ref&gt; Because our concentration is selective and focuses on discrete information, whether subjectively or objectively, we exclude other stimuli present even within our range of experience.

Our perceptions focus on what we direct our attention toward.

The spectrum of [[w:Visible_spectrum|light visible to humans]] is only a tiny fraction of the [[w:Electromagnetic_spectrum|electromagnetic spectrum]]. Radio waves, microwave radiation, infrared radiation, visible light, ultraviolet radiation, x-rays, and gamma radiation are all electromagnetic waves of various wavelengths. These various forms of radiation span wavelengths ranging from as small as 10&lt;sup&gt;-11&lt;/sup&gt; meters (0.1 Å) to as long as 1,000 meters, however only light in the range of 400 [[w:Nanometre|nm]] – 700 nm (4x10&lt;sup&gt;-7&lt;/sup&gt; to 7x10&lt;sup&gt;-7&lt;/sup&gt; meters) is visible to humans. We have no way to directly sense radio waves, infrared and ultraviolet light, x-rays, and other forms of radiation. Humans are also unable to directly sense magnetic fields.

The [[w:Hearing_range|range of human hearing]] is approximately 20 to 20,000 [[w:Hertz|Hz]], while dogs can hear sounds frequencies as high as 45 kHz and bats can hear frequencies as high as 200 kHz. Elephants can hear sounds at 14–16 Hz, while some whales can hear [[w:Infrasound|infrasound]] as low as 7 Hz.

It is estimated that dogs, in general, have an [[w:Sense_of_smell#Variability_amongst_vertebrates|olfactory sense]] (sense of smell) approximately ten thousand to a hundred thousand times more acute than a human's.

Also, we are often [[Exploring_Worldviews/What_Fish_Don’t_See|unaware of what could be most obvious]]. 

Not only does our direct perception account for only a tiny fraction of reality, much of what we do perceive is distorted.

=== Distortions: ===
Distortions of the [[w:Sense|senses]] are called [[w:Illusion|illusions]]. Although illusions distort our perception of reality, they are generally shared by most people. Illusions may occur with any of the human senses, but visual illusions (optical illusions) are the best-known and understood.

[[File:Checker shadow illusion.svg|thumb|The squares marked A and B are the same shade of gray.]]
In the [[w:Checker_shadow_illusion|checker shadow illusion]], shown here, two identically colored squares labeled “A” and “B” appear to be very different shades of gray. Reality as we perceive it may not be reality as it is.

[[Image:Fraser spiral.svg|thumb|right|225px|Fraser spiral illusion]]
In the Fraser spiral illusion, shown here, the overlapping black arc segments appear to form a spiral; however, the arcs are a series of concentric circles.

[[File:Duck-Rabbit illusion.jpg|thumb| What do you see? Are you sure?]]
In addition to illusions, many [[w:Ambiguous_image|images are ambiguous]]. As an example, please observe, and then interpret the image shown here.

What do you see? What else do you see? How certain are you of your conclusion? What do you have at stake in defending your interpretation?

Most people will see either a rabbit or a duck. If you see a rabbit, look again, and try to see a duck. Similarly, if you see a duck, look again, and try to see a rabbit.

This image, called the [[w:Rabbit-duck_illusion|rabbit-duck illusion]] is inherently [[w:Ambiguity|ambiguous]]. Someone who interprets it as representing a duck has as valid a claim to accuracy as someone who interprets this as representing a rabbit. In fact, it is neither and both. It is an ambiguous image open to at least two different yet equally valid interpretations. Similarly, a [[w:Is_the_glass_half_empty_or_half_full?|glass that is half empty]] is also half full.

In addition to optical illusions, we are susceptible to illusions that distort each of our senses. These include [[w:Auditory_illusion|auditory]], [[w:Tactile_illusion|tactile]], [[w:Time_perception#Types_of_temporal_illusions|temporal]], and [[w:Illusion#Intersensory|intersensory illusions]].

[[w:Magic_(illusion)|Magicians]] are performing artists who use illusions to entertain us. [[w:Charlatan|Charlatans]], including fakes, [[w:Mysticism|mystics]], [[w:Counterfeit|counterfeiters]],  [[w:Quackery|quacks]], [[w:Conspiracy_theory|conspiracy theorists]], and [[w:Confidence_trick|con artists]], use illusions to deceive, cheat, and swindle us. [[w:Ideology|Ideologs]] exploit illusions to promote their cause, [[w:Special_effect|special effects]] are often used in films and other artistic media to entertain us, and recently [[w:Deepfake|deepfakes]] are used to deceive us. 

In addition to our perceptions being distorted, we also sometimes perceive things which are not there. 

=== Additions: ===
Our perception systems create [[w:Perception#Process_and_terminology|percepts]] that do not otherwise exist in the world. Examples include [[w:Color_vision|color perception]], [[w:Illusory_contours|illusory contours]] and other patterns studied within [[w:Gestalt_psychology|Gestalt psychology]], [[w:Apophenia|illusory pattern recognition]], pain, phantom limbs, [[w:Hallucination|hallucinations]], and distortions that occur during various [[w:Altered_state_of_consciousness|altered states of consciousness]]. Our conceptions also influence our perceptions.

Humans [[w:Perception|perceive]] the [[w:Red|color red]] when we look at light with a wavelength between approximately 625 and 740 [[w:Nanometre|nanometers]]. Note that although electromagnetic radiation with a wavelength of approximately 700nm does exist in objective reality independent of the direct perception of any human, the percept we call red only exists in  our minds as a construct of the perceptual system. This is a consequence of our [[w:Trichromacy|trichromatic color vision]] system, including our visual cortex and [[w:Color_vision#Color_in_the_human_brain|other brain regions]]. Because color only exists as a feature constructed by the visual perception of an observer it is a [[w:Color_vision#Subjectivity_of_color_perception|subjective experience]].  When I see red, I have a subjective experience called redness. However, philosopher John Locke proposed a thought experiment, called the [[w:Inverted_spectrum|inverted spectrum]], where we imagine two people sharing their color vocabulary and discriminations, although the colors one sees—one's [[w:Qualia|qualia]]—are systematically different from the colors the other person sees. For example, perhaps the color we call red creates within me the subjective experience that the color we call green generates within you. We do not know if the subjective experience I call redness is like the subjective experience you have when seeing red.&lt;ref&gt;Siegel, Susanna, &quot;[https://plato.stanford.edu/archives/fall2021/entries/perception-contents The Contents of Perception]&quot;, The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), Edward N. Zalta (ed.), Section 4.1.&lt;/ref&gt;  The nature of redness is further explored in a philosophical thought experiment known as the [[w:Knowledge_argument|knowledge argument]], or Mary’s room.

[[Image:Kanizsa triangle.svg|thumb|right|''Kanizsa's triangle'': Do you see a white triangle?]]

This image, called [[w:Illusory_contours|Kanizsa’s triangle]], gives the impression of a bright white triangle, defined by a sharp illusory contour, occluding three black circles and a black-outlined triangle. Even knowing the bright white triangle does not exist, it is reliably constructed by our visual perception system. This is one example of many [[w:Illusory_contours|illusory contours]] that evoke the perception of an edge without a shading or color change across that edge. We see an edge where none exists. [[w:Gestalt_psychology|Gestalt psychology]] studies many such images where it appears that “The whole is more than the sum of its parts.”

[[w:Apophenia|Apophenia]] is the tendency to perceive meaningful connections between unrelated things. For example Gamblers may imagine that they see patterns in the numbers that appear in lotteries, card games, or roulette wheels, where no such patterns exist.

[[File:Martian face viking cropped.jpg|thumb|Satellite photograph of a [[w:mesa | mesa]] in the [[w:Cydonia (Mars)|Cydonia region of Mars]], often called the [[w:Cydonia (Mars)#&quot;Face on Mars&quot;|&quot;Face on Mars&quot;]] and cited as evidence of [[w:Extraterrestrial life|extraterrestrial]] habitation.]]

One form, called [[w:Pareidolia|pareidolia]], is the tendency for perception to impose a meaningful interpretation on a nebulous stimulus, usually visual, so that one sees an object, pattern, or meaning where there is none. People may mistakenly interpret an object, shape, or configuration with perceived &quot;face-like&quot; features as being a face. Many people claim to see the [[w:Man_in_the_Moon|man in the moon]]. In this satellite photograph of a [[w:mesa | mesa]] in the [[w:Cydonia (Mars)|Cydonia region of Mars]], often called the [[w:Cydonia (Mars)#&quot;Face on Mars&quot;|&quot;Face on Mars&quot;]] has been cited as evidence of [[w:Extraterrestrial life|extraterrestrial]] habitation.

As another example, what we feel as [[w:Pain|pain]] is our perception of tissue damage. Consider our perception of a [[w:Toothache|toothache]]. The damaged tooth stimulates nerve impulses in our [[w:Somatosensory_system|somatosensory nervous system]]. The brain perceives various nerve impulses transmitted through these neural structures as the unpleasant sensory and emotional experience we call pain. Notice that the brain receives only a pattern of nerve impulses and then constructs the perception of pain from those nerve impulses. Although the damage is in the tooth the perception of pain is constructed in the brain.

A [[w:Phantom_limb|phantom limb]] is the sensation that an amputated or missing limb is still attached. Approximately 80 to 100% of individuals with an amputation experience sensations in their amputated limb. In phantom limb syndrome, there is sensory input indicating pain from a part of the body that no longer exists. This phenomenon is still not fully understood, but it is hypothesized that it is caused by activation of the [[w:Somatosensory_cortex|somatosensory cortex]]. The perception of the absent limb is constructed by our nervous system.

As another example, [[w:Tinnitus|tinnitus]] is the perception of sound when no corresponding external sound is present.

Our perceptions are often distorted when we are experiencing [[w:Altered_state_of_consciousness|altered states of consciousness]]. This may occur from the influences of drugs—especially [[w:Psychoactive_drug|psychoactive drugs]] including [[w:Hallucinogen|hallucinogens]] and [[w:Alcohol_(drug)|alcohol]]—fatigue, disease, [[w:Hypoxia_(medical)|hypoxia]] (as can occur from [[w:Hypoventilation|hypoventilation]] and other [[w:Breathing|breath]] control practices), [[w:Hallucination|hallucinations]], [[w:Delirium|delirium]], and various mental illnesses. 

Our conceptions often influence our perceptions. Try this simple quiz:
&lt;blockquote&gt;
What does F-O-L-K spell? (Please say the word out loud.)&lt;br&gt;
What is the white of an egg called?  (Please say the word out loud.)&lt;br&gt;
(What color is the white of an egg?)
&lt;/blockquote&gt;
If you said “yolk”, as many people do, you were influenced by psychological priming. [[w:Priming_(psychology)|Priming]] is a phenomenon whereby exposure to one stimulus influences a response to a subsequent stimulus, without conscious guidance or intention. For example, in experiments the word ''nurse'' is recognized more quickly following the word ''doctor'' than following the word ''bread''.

We may be unaware of priming effects that arise [[w:Priming_(psychology)#In_daily_life|in our daily lives]]. In one study subjects were implicitly primed with words related to the stereotype of elderly people (example: Florida, forgetful, wrinkle). While the words did not explicitly mention speed or slowness, those who were primed with these words walked more slowly upon exiting the testing booth than those who were primed with neutral stimuli.

There is more to perception than meets the eye!

=== Assisted Perception—Compensating for the quirks ===
Fortunately, there are many methods we can use to overcome and compensate for the deficiencies, distortions, quirks, and other characteristics of our direct perception systems. Microscopes, telescopes, [[w:Oscilloscope|oscilloscopes]], many other [[w:Scientific_instrument|scientific instruments]], recording devices, and [[w:Technical_standard|reference standards]] allow us to extend the reach, scope, and accuracy of our observations. Multiple observers, vantage points, perspectives, and viewpoints allow us to gain a more complete and reliable examination of reality when we share and integrate information. 

Applying the principle of [[w:Consilience|consilience]] and [[Thinking Scientifically|thinking scientifically]] increase the reliability of our observations and can help us see beyond illusions. 

==== Assignment ====
#Observe the moon in the sky some evening.
#Estimate the size (diameter) of the moon.
#Estimate the distance the moon is from you.
#Research the [[w:Moon|accepted values of these numbers]] and compare them to your direct observations. How closely does the perceived distance and size compare to the accepted values?
#Optionally repeat this for the sun, bright stars, and other distant terrestrial and celestial objects. 

=== Perceptions are Personal ===
We often hear that “perception is reality” and that “everything is relative”, despite knowing that a shared reality exists, and reality is our common ground.

Perceptions are vivid. Seeing things from our own point of view is always easier, and first-hand experiences seem more real than understanding another's point of view can ever be. Our eyes, nose, taste buds, tactile sensors, and ears connect directly only to our brain. Only you experience first-hand the direct sensory input of the world; you, your self, is the observer. This raw sensory input is interpreted and gains meaning through your unique perceptions and past experiences. Furthermore, contemplation, desire, intent, pain, introspection, consciousness, and reflection are all private and solitary. This unique first-person experience creates a fundamental asymmetry that contributes to many of the other asymmetries that govern social interactions. It also contributes to the asymmetric character of [[Coping with Ego|egotism]], narcissism, selfishness, greed, and the magnitude gap. We judge others based on behavior and we judge ourselves based on intent. Your own point of view, the way you see things, is unique. The [[Living the Golden Rule|golden rule]] and our empathy struggle to overcome this fundamental imbalance.

It is often a [[w:Problem_of_induction|mistake to generalize]] our personal perceptions beyond our own experiences. Standing in a meadow we see a flat earth, yet sunrise, time zones, global travel, earth satellites, GPS navigation systems, images from space, and travel to the moon all assure us the [[w:Spherical_Earth|earth is nearly spherical]]. Reality is vast, complex, and dynamic, and our perceptions are only a tiny glimpse of all there is to know about reality. Only a [[Global Perspective|global perspective]] brings us uncensored reality.

Reality exists and provides us with [[Facing_Facts|matters of fact]]. The [[Knowing_How_You_Know/Height_of_the_Eiffel_Tower|Eiffel tower is 300 meters tall]]. You may perceive that as too short; others may perceive that as too tall, and many perceive that as simply beautiful.
 
See beyond the illusion that what you see is all there is. Perceptions are personal, but [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]]. Our unique first-person viewpoint creates a powerful asymmetry that requires deliberate effort to see beyond. Transcend your personal perceptions, investigate the vastness and complexity of reality from a [[Global Perspective|global perspective]], and embrace reality as our common ground.
==== Assignment ====
#Read the essay [[Facing_Facts/Perceptions_are_Personal|Perceptions are Personal]].
#Reflect on occasions when you projected your own perceptions onto others or beyond your own experience.
#Recognize the limited scope and likely deficiencies of your perceptions.

== Representation ==
At least until we become proficient at [[w:Telepathy|mental telepathy]], we need some other way to represent our thoughts and to communicate our thoughts to others.

[[w:Word|Words]] are one type of [[w:Symbol|symbol]] used to represent reality. Other symbols might include [[w:Vocable|sounds]], [[w:List_of_gestures|gestures]], [[w:Facial_expression|facial expressions]], [[w:Icon_(computing)|icons]], [[w:Logo|logos]], [[w:Drawing|drawings]], [[w:Model|models]], [[w:Analogy|analogies]], [[w:Metaphor|metaphors]], and other representations. The mapping from words to perceptions to reality [[w:Polysemy|is often imprecise]]. We may describe a certain color simply as ''red'', without clarifying what [[w:Shades_of_red|shade of red]] we are describing. If I tell you I am sitting in a chair, you might imagine a rocking chair, a desk chair, or a recliner. [[w:Syntactic_ambiguity|Syntactic ambiguity]] often adds to the confusion.

Do not mistake the symbol for reality. The [[w:Noumenon|noumenal]] world exists independent of our senses and may differ from the [[w:Noumenon|phenomenal]] world we perceive and the symbols we use to represent it. 

Language and other symbols can be constraining, ambiguous, refer to social constructs, prejudiced, based on artificial boundaries, or focus on convenient representation rather than reality. These features, distortions, and inaccuracies are explored further below.

=== Constraining Language ===
[[File:Clip.jpg| thumb|right|What can you do with this wire form? What would you call it?]]
Consider the image on the right. What can we use it for? This wire form is commonly called a [[w:Paper_clip|paper clip]] and is typically used to clip together sheets of paper. However, [[Thinking_Tools#Lateral_Thinking|lateral thinkers]] have identified at least one [[w:Paper_clip#Other_uses|hundred alternative]] uses&lt;ref&gt;100 Uses for Paperclips, See: https://leoniehallatinnovationiq.wordpress.com/2012/11/21/100-uses-for-paperclips/ &lt;/ref&gt;  for this object. Observing the object, thinking expansively and creatively about what it might be, how it can be used, or what it might do, can identify many possibilities. This unclassified reality invites many potential uses. Once the object is named, an interpretation is imposed, and the object becomes restricted.

Naming the object assigns the object to one category, as it excludes others. 

This table describes possibilities before and after it is interpreted, assigned to a category, and named.

{| class=&quot;wikitable&quot;
|+ Categorization Constrains Imagination
|-
! Encounter with reality: !! Interpretation and Representation:
|-
|
*An experience
*Awareness
*Curious
*Exploration
*Investigation
*Witness
*Imagine
*Possibilities
*Potential
*Opportunities
*An affordance
*Lateral Thinking
*No (one) thing
*The territory
*Could be …
||
*Analogy
*Categories
*Labels
*Restrictions
*Evaluation
*Judgement
*Purpose
*Specifics
*''This'' map
*''This'' thing
**A paperclip
**A lock pick
**A hook
**DVD drive opener
**Earrings
|}
	
Language and other symbols are ambiguous, persuasive, and subtle, and can be used heroically, precisely, expansively, restrictively, deceptively, and manipulatively. Here are some examples.

=== Ambiguous Language ===
[[w:Polysemy|'''Polysemy''']] is the capacity for a sign (e.g., a symbol, a word, or a phrase) to have multiple related meanings. A word can have several [[w:Polysemy|word senses]]. As an example, the word ''bank'' can have at least these 7 distinct meanings:
#a financial institution
#the physical building where a financial institution offers services
#to deposit money or have an account in a bank (e.g., &quot;I bank at the local credit union&quot;)
#[[w:Bank_(geography)|a steep slope]] (as of a hill or the rising ground bordering water)
#[[w:Banked_turn#Banked_turn_in_aeronautics|to incline an airplane]] laterally
#a supply of something held in reserve: such as &quot;banking&quot; brownie points
#a synonym for 'rely upon' (e.g. &quot;I'm your friend, you can bank on me&quot;).

[[w:Syntactic_ambiguity|'''Syntactic ambiguity''']] is language where a [[w:Sentence_(linguistics)|sentence]] may be interpreted in more than one way due to ambiguous [[w:Syntax|sentence structure]]. For example, the sentence ''John saw the man on the mountain with a telescope'' can have these various interpretations. 
*John, using a telescope, saw a man on a mountain.
*John saw a man on a mountain which had a telescope on it.
*John saw a man on a mountain who had a telescope.
*John, on a mountain and using a telescope, saw a man.
*John, on a mountain, saw a man who had a telescope.

Combining polysemy with syntactic ambiguity results in multiplying the ambiguity. For example, Gerald Weinberg identifies dozens of interpretations of the sentence “Mary had a little lamb”&lt;ref&gt; {{cite book |last1=Gause  |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is  |publisher=Dorset House Publishing Company |pages=176 |isbn=978-0932633163 |author-link=w:Gerald_Weinberg}} @128 of 255.&lt;/ref&gt;  and then goes on to suggest 20 more techniques for identifying ambiguity in sentences.

[[Exploring_Social_Constructs#Reifications|'''Reifications''']] are a class of nouns that do not refer to real objects but rather to abstract concepts. Examples include justice, freedom, liberty, equality, rights, duty, responsibility, truth, fairness, and citizen and extend to include concepts such as race, land ownership, owning coal, money, debt, contracts, and even agreement. These labels cannot be resolved to [[w:Brute_fact#Searle|brute facts]] but are often treated as if they do. This results in a [[w:Reification_(fallacy)|reification fallacy]]. A [[w:Map–territory_relation#&quot;A_map_is_not_the_territory&quot;|map is not the territory]], an [[w:The_Treachery_of_Images|image of a pipe is not a pipe]], and there is little or no agreement on what “[[w:justice|justice]]” is. 

Simple phrases such as “we seek justice”, “justice was served”, “[[w:Pledge_of_Allegiance|liberty and justice for all]]”, “[[w:Preamble_to_the_United_States_Constitution|establish justice]]”, and “ours is a nation of laws” become very ambiguous, complex, and controversial when we recognize the wide range of meanings attributed to the abstract concept of “justice”.

The concept of [[w:justice|justice]] differs in every culture. Early theories of justice were set out by the Ancient Greek philosophers Plato in his work [[w:Republic_(Plato)|The Republic]], and Aristotle in his [[w:Nicomachean_Ethics|Nicomachean Ethics]]. Throughout history various theories have been established. Advocates of [[w:Divine_command_theory|divine command theory]] argue that justice issues from God. In the 1600s, theorists like [[w:John_Locke|John Locke]] argued for the theory of [[w:Natural_law|natural law]]. Thinkers in the [[w:Social_contract|social contract]] tradition argued that justice is derived from the mutual agreement of everyone concerned. In the 1800s, [[w:Utilitarian|utilitarian]] thinkers including [[w:John_Stuart_Mill|John Stuart Mill]] argued that justice is what has the best consequences. Theories of [[w:Distributive_justice|distributive justice]] concern what is distributed, between whom assets or liabilities are to be distributed, and what is the ''proper'' distribution. [[w:Egalitarianism|Egalitarians]] argued that justice can only exist within the coordinates of equality. [[w:John_Rawls|John Rawls]] used a [[w:Social_contract|social contract]] argument to show that justice, and especially distributive justice, is a form of [[Understanding Fairness|fairness]]. Property rights theorists (like [[w:Robert_Nozick|Robert Nozick]]) also take a consequentialist view of distributive justice and argue that property rights-based justice maximizes the overall wealth of an economic system. Theories of retributive justice are concerned with [[w:Punishment|punishment]] for wrongdoing. [[w:Restorative_justice|Restorative justice]] (also sometimes called &quot;reparative justice&quot;) is an approach to justice that focuses on the needs of victims and offenders.

There are many [[w:Wikipedia:Manual_of_Style/Words_to_watch|words to avoid]] when trying to be objective and precise. 
===== Assignment =====
#Choose a speech or other text to use for this assignment. This may be chosen from this [[w:List_of_speeches|list of speeches]], the Wikisource [[s:Portal:Speeches|speeches portal]], [[w:List_of_amendments_to_the_United_States_Constitution|amendments to the United States Constitution]], the [[w:Universal_Declaration_of_Human_Rights|Universal Declaration of Human Rights]], [[w:List_of_landmark_court_decisions_in_the_United_States|landmark court decisions]], or any other source.
#Identify instances of ambiguous language in the chosen text.
#For at least three instances of ambiguous language, list various meanings that can be reasonably implied by the language used. For example, the ambiguous phrase “in the future” could mean seconds, hours, days, weeks, months, years, centuries, or eons from now. 
#Suggest more precise and objective language that could be substituted for the language used. 
#Suggest an [[w:Operational_definition|operational definition]] to replace or clarify the ambiguous language. 
==== Social Constructs ====
When we endow [[w:Brute_fact#Searle|''brute facts'']] with additional status, we create [[Exploring_Social_Constructs|''social constructs'']].

Social constructs rely on collective human agreement, or at least acceptance of the imposition of the new status Y on the brute fact X.

Social constructs include games such as soccer, baseball, and chess. Bureaucracies including clubs, organizations, and corporations are social constructs. Titles such as chairman, president, king, and pope, along with governments, financial instruments, property ownership agreements, and religions are social constructs.

Social constructs are [[Exploring_Social_Constructs#Ambiguity|ambiguous]], sometimes fragile, and often require [[Exploring_Social_Constructs#Referees|referees]] of some form. The agreements used to form the social constructs may be obsolete or challenged. Social constructs can be [[Exploring_Social_Constructs#Mismatches|mismatched]] to relevant brute facts.

Because social constructs are so common and so prominent, we can easily mistake them for brute facts. This is an error. Mistaking social constructs for brute facts introduces several layers of abstraction and creates distortions that distance us from realty. 

As a result of ambiguity and defective agreements many social constructs are poorly aligned with the brute facts they are based on, and many mismatches occur. These cause friction in our society and can contribute to [[Grand challenges|many challenges]] we face. By observing the mismatch of social constructs to brute facts and informed consent, we can begin to troubleshoot and improve the collection of social constructs that create our culture and institutions.

The bad news is that we face many issues resulting from social constructs misaligned with brute facts or based on defective agreements. The good news is that because social constructs are human constructs, we can work to improve them.
===== Assignment =====
#Choose a speech or other text to use for this assignment. This may be chosen from this [[w:List_of_speeches|list of speeches]], the Wikisource [[s:Portal:Speeches|speeches portal]], [[w:List_of_amendments_to_the_United_States_Constitution|amendments to the United States Constitution]], the [[w:Universal_Declaration_of_Human_Rights|Universal Declaration of Human Rights]], [[w:List_of_landmark_court_decisions_in_the_United_States|landmark court decisions]], or any other source.
#Identify instances of ''social constructs'' that appear in the chosen text.
#For at least three instances of the social constructs identified, list various meanings that can be reasonably implied by the language used. For example, the [[w:Second_Amendment_to_the_United_States_Constitution|second amendment]] uses the term “[[w:Militia|Militia]]”.  This could mean military unit, police units, paramilitary units, security guards, or individuals. 
#Suggest more precise and objective language that could be substituted for the language used. 
#Suggest an [[w:Operational_definition|operational definition]] to replace or clarify the ambiguous language. 

==== Prejudiced Language ====
[[w:Loaded_language|Loaded language]] is [[w:Rhetoric|rhetoric]] used to influence an audience by using words and phrases with strong [[w:Connotations|connotations]]. This type of language is unclear (vague) and can be used to [[w:Pathos|invoke an emotional response]] or exploit [[w:Stereotypes|stereotypes]].  Loaded words and phrases have significant emotional implications and involve strongly positive or negative reactions beyond their [[w:Literal_meaning|literal meaning]]. These may include [[w:Racism|racists]] or [[w:Sexism|sexist]] words, [[w:List_of_ethnic_slurs|ethnic slurs]], [[w:List_of_religious_slurs|religious slurs]], and other [[w:Lists_of_pejorative_terms_for_people|pejorative terms]]. 

[[w:Wikipedia:Manual_of_Style/Words_to_watch#Contentious_labels|Examples of contentious labels include]]: cult, racist, perverted, sexist, homophobic, transphobic, misogynistic, sect, fundamentalist, heretic, extremist, denialist, terrorist, freedom fighter, bigot, myth, neo-Nazi, -gate, pseudo-, controversial, and others.
==== Beware of Boundaries ====
[[File:Tannin heap.jpeg|thumb|The [[w:Sorites_paradox|sorites paradox]]: If a heap is reduced by a single grain at a time, at what exact point does it cease to be considered a heap?]]

The [[w:Sorites_paradox|sorites paradox]] poses the question, if removing one grain from a heap of sand leaves it a heap, then one grain of sand is also a heap. When does a heap of sand transform into a few grains of sand that are no longer a heap? This paradox illustrates that the concept of ''heap'' is ambiguous. Also, the boundary between a heap and some smaller collection is also ambiguous. 

In conventional language, logic, mathematics, and decision-making, we generally regard [[Natural_Inclusion/Boundaries|boundaries]] as discrete or definitive limits or borders, which permanently and absolutely divide one thing or locality apart from other things or localities. For such definitive limits to exist, however, they would have to be so sharp as to have no thickness. By contrast, even when viewed from afar and over short durations, natural boundaries often appear diffuse, mobile, and impermanent, defying such precise, abstract definition. Naturally occurring boundaries are inherently ambiguous. Artificial boundaries are often sharply defined. This mismatch between how boundaries occur naturally and how we chose to represent them can lead to several problems. 

Racial classifications create problems because they impose sharply defined boundaries where no natural boundary exists. [[w:Race_(human_categorization)|Racial classifications]] are [[Exploring_Social_Constructs|socially constructed]]. While partially based on physical similarities within groups, race does not have an inherent physical or biological meaning. Therefore, race assignment is inherently ambiguous. None-the less, laws prevail that impose harsh burdens based on racial classification. These laws define sharp boundaries to separate one race from another. Simplistic resolutions of racial ambiguity are the [[w:One-drop_rule|one-drop rules]] that asserted that any person with even one ancestor of black ancestry ('one drop' of 'black blood') is considered black. Attempts to define [[w:Native_American_identity_in_the_United_States#Blood_quantum|Native American identity in the United States]] encounters similar difficulties.

==== The map is not the territory. ==== 
[[File:Azimuthal equidistant projection with Tissot's indicatrix.png|thumb|A map is not an accurate representation of the territory it seeks to illustrate— in this example, the increasing distortion is represented by the circles gradually becoming ellipses]]
As soon as we label an object to represent it, use a [[w:Mental_model|mental model]], invoke an [[w:Analogy|analogy]], use a [[w:Metaphor|metaphor]], substitute a representation for a thought, idea, or object, or substitute an interpretation for some set of observations we substitute [[w:Map–territory_relation|the map for the territory]].

Our brain learns a model of the world. Intelligence is tied to the model the brain creates. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}@22 of 384&lt;/ref&gt;  The brain learns its model of the world by observing how its inputs change over time. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}} @66 of 384&lt;/ref&gt; The neocortex learns a predictive model of the world. &lt;ref&gt;{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}@75 of 384&lt;/ref&gt;

We rely on the map our brain creates to navigate the world we live in. This works best when our mental maps correspond accurately to the real world. We navigate through life guided by the mental representations that form in our brains. If you are now sitting on a chair in a room, and you want to leave that room, you rely on your mental image of the chair, your location in the room, and the location of the door. If this mental model is wrong, you will have difficulty getting out of the chair, walking across the room, opening the door, and walking through.

We know [[w:Map–territory relation|the map is not the territory]], but merely some simplified and often distorted representation of the corresponding territory—reality as it is. Whenever we rely on the map rather than the territory for forming beliefs, deciding, or planning actions, we risk misrepresenting the territory and embracing a distorted view of reality.

George Box reminds us that “[[w:All_models_are_wrong|All models are wrong]], some are useful”.

We depart from reality the moment we move from [[w:Perception|''perception'']] to [[w:Concept|''conception'']] from [[w:Observation|''observation'']] to [[w:Interpretation_(philosophy)|''interpretation'']].  In his painting [[w:The_Treachery_of_Images|''The Treachery of Images'']], [[w:René_Magritte|René Magritte]] reminds us that an image of a pipe is a representation of the pipe and not the pipe itself.

It is difficult to be precise and neutral in our language. Language provides many opportunities to be vague, ambiguous, [[w:Bullshit|nonsensical]], prejudicial, emotional, laudatory, persuasive, kind, cruel, [[w:Buzzword|vacuous]], or [[w:Weasel_word|equivocal]]. Because [[w:Rhetoric|rhetoric]] is the art of persuasion, be aware that it can draw you toward a conclusion and influence your beliefs using only baseless arguments and emotional manipulation.

=== Assignment ===
#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of language, perception, or reality.
#[[Practicing Dialogue|Practice dialogue]] rather than debate or argumentation. Be candid. [[Living_Wisely/Advance_no_falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course.
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[Evaluating_Evidence|research them]].
##[[Seeking True Beliefs|Seek true beliefs]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Interpretation ==
People [[w:Interpretation_(philosophy)|interpret]] observations to resolve ambiguity, increase certainty, and to account for observations within some familiar model or analogy. The role of interpretation is most evident when the items being interpreted are most [[w:Ambiguity|ambiguous]]. 
=== Tea Leaves ===
[[File:Spring pouchong tea leaves on plate.jpg|thumb|''Spring Pouchong'' tea leaves that may be used for [[w:Tasseography|tasseography]] divination]]
[[w:Tasseography|Tasseography]] is the art of interpreting patterns in tea leaves, coffee grounds, or wine sediments. The diviner—a person skilled in interpreting tea leaves—looks at the pattern of tea leaves in the cup and allows the imagination to play around with the shapes suggested by them. They might look like a letter, a heart shape, a ring, or anything else. These shapes are then interpreted intuitively or by means of some system of symbolism. Images formed in a cup are created and uniquely seen by the reader, so it is often said that the only limitation for cup reading is the imagination of the reader themselves.

Tasseography reveals the essence of interpretation at an extreme. Because the tea leaf formations are entirely arbitrary the interpreter can say almost anything. The interpretation reflects judgements, evaluations, biases, concepts, models, analogies, predictions, optimism, pessimism, certainty, risk, doubt, hope, fears, storytelling skills, and desires of the interpreter, entirely independent of the tea leaf formation.

[[w:Rorschach_test|Rorschach tests]], [[w:Astrology|astrology]], [[w:Horoscope|horoscopes]], [[w:Biorhythm_(pseudoscience)|biorhythms]], [[w:Tarot_card_reading|tarot card reading]], [[w:Ouija|Ouija]], [[w:Fortune-telling|fortune telling]], and [[w:Dream_interpretation|dream interpretations]] are similar to tasseography in that they all depend more on the person doing the interpretation than on the ambiguous items being interpreted.

Recognizing the extent of ambiguity inherent in the various representations we use to describe reality, interpretation plays a significant role in influencing our understanding and beliefs. Adopting an interpretation can obscure and nearly occlude the underlying reality. Keep your eye on the territory as others offer various maps they would like you to use instead. Continue to reference reality so you can evaluate, challenge, and often reject, various interpretations.

=== Reality, Perception, and Interpretation ===
[[File:Blind monks examining an elephant.jpg|thumb|A depiction of the [[w:Blind_men_and_an_elephant|blind men and the elephant]]]]
The parable of the [[w:Blind_men_and_an_elephant|blind men and the elephant]] provides an example that can help us examine the distinctions among [[w:Reality|''reality'']], [[w:Perception|''perception'']], and [[w:Interpretation_(philosophy)|''interpretation'']]. In the parable is a story of a group of blind men who have never come across an elephant before and who learn and imagine what the elephant is like by touching it. Each blind man feels a different part of the elephant's body, but only one part, such as the tail or the tusk. They then describe the elephant based on their limited experience and their descriptions of the elephant are different from each other.

The elephant is an example of ''reality''—what exists in the world. Each of the blind men makes some limited ''observations'' and forms a ''perception'' of the elephant. Each of these perceptions is based a small sample of reality. The man at the tail accurately perceives the tail by sensing how it feels. The man at the tusk accurately perceives the tusk, also by sensing how it feels. Then each man ''interprets'' his (limited) observation. The man at the tail concludes an elephant is like a rope, the man at the tusk concludes the elephant is hard, smooth, and like a spear.

Each blind man makes errors when interpreting their perceptions:
*Each fails to include the observation of the others, and if they consult the other observers, they fail to trust these additional observations and integrate them into a consistent whole. 
*Each [[w:Faulty_generalization|overgeneralizes]] from their limited experience to draw conclusions about the entire elephant.
*Each interprets their observations in terms of existing [[w:Paradigm|paradigms]] (rope, spear) rather than considering new paradigms (elephant). 

Recall the differing yet equally correct interpretations of the [[w:Rabbit–duck illusion|rabbit-duck illusion]].  This simple image allows for two different, yet equally correct interpretations. Life is often more complicated than ducks and rabbits, and we can reasonably differ in interpreting many words.

The [[w:Second_Amendment_to_the_United_States_Constitution|second amendment to the United States Constitution]] is the subject of long and contentious interpretations. It states:
&lt;blockquote&gt;
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
&lt;/blockquote&gt;
The (sometimes omitted) commas in the text draw much attention. The Militia clause is especially open to interpretation. Indeed, most of the words in this short statement have been interpreted in a variety of ways. Interpretation of this text is the subject of [[w:List_of_firearm_court_cases_in_the_United_States|many court cases]] and continues to sharply divide political discourse in the United States.

[[w:Innuendo|Innuendo]] and [[w:Plausible_deniability|plausible deniability]] [[w:Deception|disingenuously]] [[w:Equivocation|equivocate]] on interpretation.

Learn to [[Embracing_Ambiguity|embrace ambiguity]]. [[Finding_Common_Ground/Doubt_and_our_Bayesian_Brains|Become comfortable with doubt]]. Examine and reevaluate your preconceptions. Avoid tragic misinterpretations.

=== Tragic Interpretations ===
Interpretations that prematurely eliminate [[w:Doubt|doubt]] and resolve [[w:Ambiguity|ambiguity]] into a comfortable yet false feeling of [[w:Certainty|certainty]] have led to several tragedies. Here are some prominent examples.
#It has long been observed that the sky brightens each morning and day turn to darkness each evening. The causes of this were ambiguous for most of recorded history. Perhaps the sun moves around the earth on a [[w:Celestial_sphere|celestial sphere]]. Alternatively, the earth could rotate on its axis as it [[w:Heliocentrism|revolves around the sun]]. The pope was certain the earth was the center of the universe, [[w:Galileo_Galilei|Galileo]] differed. The contentious [[w:Galileo_affair|debate over these alternative interpretations]] of the observations culminated with the trial and condemnation of [[w:Galileo_Galilei|Galileo Galilei]] by the [[w:Roman_Inquisition|Roman Catholic Inquisition]] in 1633.
#In 1997 members of the [[w:Heaven's_Gate_(religious_group)#Mass_suicide|Heaven’s Gate]] new religious movement misinterpreted images of the Hale-Bopp comet, and decided that the only way to evacuate this earth was to participate in a mass suicide. [[w:Cult|Cults]] are often formed based on alternative interpretations of events.
#Although the phrase “[[w:All_men_are_created_equal|All men are created equal]]” motivated [[w:American_Revolutionary_War|the revolution]] that formed the United States, differing interpretations of the status of [[w:Slavery_in_the_United_States|slaves]] as being either property or being humans led to the [[w:American_Civil_War|civil war]]. 
#Religious groups often differ in the interpretation of various symbols, texts, and prophesies. Here are some examples.
##[[w:Christianity|Christianity]] is a religion based on interpretations of the [[w:Life_of_Jesus_in_the_New_Testament|life]] and [[w:Teachings_of_Jesus|teachings]] of [[w:Jesus|Jesus of Nazareth]]. These interpretations lead to the belief that [[w:Jesus|Jesus]] is the [[w:Son_of_God_(Christianity)|Son of God]], whose coming as the [[w:Messiah#Christianity|messiah]] was [[w:Old_Testament_messianic_prophecies_quoted_in_the_New_Testament|prophesied]] in the [[w:Hebrew_Bible|Hebrew Bible]] and chronicled in the [[w:New_Testament|New Testament]].
##The [[w:Judaism#Christianity_and_Judaism|differences between Christianity and Judaism]] originally centered on whether Jesus was the Jewish Messiah but eventually became irreconcilable. Followers of [[w:Judaism|Judaism]] interpret the life of Jesus to be that of a well-meaning and charismatic human who worked as carpenter, but not the Messiah. This difference in interpretation led to the [[w:The_Holocaust|holocaust]]. 
##[[w:Islam|Islam]] is a religion teaching that [[w:Muhammad|Muhammad is a messenger of God]]. The primary scriptures of Islam are the [[w:Quran|Quran]], interpreted as the verbatim word of God. [[w:Islam#Denominations|Various denominations]] differ in their interpretations of the rightful successors of Muhammad. This difference in interpretation has led to [[w:Human_rights_in_post-invasion_Iraq#Sectarian_warfare_in_Iraq|sectarian warfare]]. 
##[[w:Scientology|Scientology]], [[w:Scientology#Scientology_as_a_religion|classified as a religion]] by the United States Internal Revenue Service, is a set of beliefs and practices invented by American author [[w:L._Ron_Hubbard|L. Ron Hubbard]], and an associated movement. It has been variously defined as a [[w:Cult|cult]], a [[w:Scientology_as_a_business|business]], or a [[w:New_religious_movement|new religious movement]], depending on various interpretations of the various [[w:Scientology_beliefs_and_practices|beliefs and practices]].
##[[w:Nontheism|Nontheists]] study reality and recognize that [[Beyond_Theism#Non-Theism_is_the_Null_Hypothesis|non-theism is the parsimonious worldview]]. Therefore, theists who make supernatural claims bear the (unmet) burden of proving their supernatural claims. Nontheists may risk persecution for [[w:Blasphemy|blasphemy]]. 
##[[w:Religious_war|Religious wars]], often resulting from disputes over these various interpretations, are frequent, long lasting, and deadly. Matthew White's [[w:The_Great_Big_Book_of_Horrible_Things|''The Great Big Book of Horrible Things'']] gives religion as the primary cause of 11 of the world's 100 deadliest atrocities.
#The nature of the [[w:2021_United_States_Capitol_attack|2021 United States Capitol attack]] has been widely, passionately, and [[w:Domestic_reactions_to_the_2021_United_States_Capitol_attack|variously interpreted]]. Former attorney general [[w:William_Barr|William Barr]], who had resigned days earlier, denounced the violence, calling it &quot;outrageous and despicable&quot;, adding that the president's actions were a &quot;betrayal of his office and supporters&quot; and that &quot;orchestrating a mob to pressure Congress is inexcusable.&quot; None-the-less, the Republican National Committee contended that the lethal riot was an example of &quot;legitimate political discourse.&quot; [[w:2021_United_States_Capitol_attack#Aftermath|The aftermath]] continues to have important political, legal, and social repercussions.
#Various [[w:Conspiracy_theory|conspiracy theories]] are based on alternative interpretations of events. Here are a few selected from a much longer [[w:List_of_conspiracy_theories|list of conspiracy theories]].
##Many [[w:John_F._Kennedy_assassination_conspiracy_theories|conspiracy theories concerning the assassination of John F. Kennedy]] in 1963 have emerged. Many of these depend on various interpretations of the available evidence and are especially skeptical of the [[w:Single-bullet_theory|single bullet theory]]—the official description of the event appearing in the [[w:Warren_Commission|Warren commission report]]. It is also frequently asserted that the United States federal government intentionally covered up crucial information in the aftermath of the assassination to prevent the conspiracy from being discovered.
##The [[w:September_11_attacks|multiple attacks]] made on the US by [[w:Terrorism|terrorists]] using hijacked aircraft on September 11, 2001 have proven [[w:9/11 conspiracy theories|attractive to conspiracy theorists]]. Theories may include reference to missile or hologram technology. By far, the most common theory is that the attacks were in fact controlled demolitions, a theory which has been rejected by the engineering profession and the [[w:9/11_Commission|9/11 Commission]].
##The &quot;[[w:Deep_state_in_the_United_States|Deep state]]&quot; often refers to discredited allegations of an unidentified &quot;powerful elite&quot; who act in coordinated manipulation of a nation's politics and government. Proponents of such theories have included Canadian author [[w:Peter_Dale_Scott|Peter Dale Scott]], who has promoted the idea in the US since at least the 1990s, as well as [[w:Breitbart_News|''Breitbart News'']], [[w:Infowars|''Infowars'']] and former US President [[w:List_of_conspiracy_theories_promoted_by_Donald_Trump|Donald Trump]]. A 2017 poll by [[w:ABC_News|ABC News]] and The Washington Post indicated that 48% of Americans believe in the existence of a conspiratorial &quot;deep state&quot; in the US. Some of these theories promote [[w:QAnon|QAnon conspiracy theories]] which are based on the interpretation of false claims made by an anonymous individual or individuals known as &quot;Q&quot;.
##[[w:Anti-vaccination|Anti-vaccination activists]] and other people in many countries have spread a variety of unfounded [[w:COVID-19_vaccine_misinformation_and_hesitancy|conspiracy theories]] and other [[w:Misinformation|misinformation]] about [[w:COVID-19_vaccine|COVID-19 vaccines]] based on misinterpreted or misrepresented science, religion, exaggerated claims about side effects, a story about COVID-19 being spread by [[w:COVID-19_misinformation#5G_mobile-phone_networks|5G]], misrepresentations about how the immune system works and when and how COVID-19 vaccines are made, and other false or distorted information. This has prolonged the pandemic and caused political unrest. 
#[[w:Quackery|Quackery]], [[w:Quackery|crystal healing]], [[w:Homeopathy|homeopathy]], and other ineffective and fraudulent health claims waste time and money while deceiving patients and delaying effective treatments. These are sustained by inaccurate interpretation of evidence.
=== Assignment ===
#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of interpretations, language, perception, or reality.
##Identify the specific interpretations that differ.
##Identify the ambiguities that allow for various interpretations.
##Cast doubt on the certainty of any specific interpretation. 
#[[Practicing Dialogue|Practice dialogue]] rather than debate or argumentation. Be [[Candor|candid]]. [[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course.
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[w:Evaluating_Evidence|research them]].
##[[Embracing Ambiguity|Embrace ambiguity]].
##[[Seeking_True_Beliefs|Seek true beliefs]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Narration ==
[[File:John Everett Millais (1829-1896) - The Boyhood of Raleigh - N01691 - National Gallery.jpg|thumb|We are captivated by [[w:Storytelling|storytelling]].]]
Humans enjoy telling and retelling [[w:Narrative|stories]]. [[w:Myth|Myths]] have been a part of human culture for at least as long as recorded history. [[w:Folklore|Folklore]], [[w:Oral_tradition|oral traditions]], [[w:Epic_poetry|epics]], [[w:Creation_myth|creation myths]], [[w:Campfire_story|campfire stories]], [[w:Fairy_tale|fairy tales]], [[w:Legend|legends]], [[w:Bedtime_story|bedtime stories]], [[w:Song|songs]], [[w:Poetry|poems]], and [[w:Soap_opera|soap operas]] are told and retold. Stories provide a memorable, coherent, compelling, and plausible explanation for events. They are also often fanciful and factually unfounded.

In his book [[w:Sapiens:_A_Brief_History_of_Humankind|''Sapiens: A Brief History Of Humankind'']], author Yuval Harari claims that all large-scale human cooperation systems – including [[w:Religions|religions]], [[w:Political_structures|political structures]], [[w:Trade|trade networks]], and [[w:Legal_institutions|legal institutions]] – owe their emergence to Sapiens' distinctive cognitive capacity for [[w:Fiction|fiction]]. Ours is the storytelling species.

=== Narratives that stick ===
Why is it that some ideas survive while others die? According to the book [[w:Made_to_Stick|''Made to Stick'']], an idea becomes memorable or interesting when it is: 
*Simple – find the core of any idea or thoughts
*Unexpected – grab people's attention by surprising them
*Concrete – make sure an idea can be grasped and remembered later
*Credible – give an idea believability and credibility
*Emotional – help people see the importance of an idea
*Presented as a story – empower people to use an idea through narrative

Several grand narratives currently divide our [[Exploring Worldviews|worldviews]] and political discourse in the United States. [[w:Conservatism|Conservatives]] often agree with Ronald Regan that “[[w:Ronald_Reagan#First_Inaugural_address_(1981)|Government is the Problem]]”, while [[w:Progressivism|progressives]] tend to believe that “Government is the solution”. Each [[w:Tribe|tribe]] can cite many examples bolstering their position. Beginning with their chosen narrative, conservatives often [[w:Mask_refusal|oppose mask mandates]], while progressives blame the unmasked for endangering others and prolonging the pandemic. These arguments begin with narratives and extend to interpretations and representative symbols but are rarely based on a careful [[Evaluating_Evidence|evaluation of evidence]].

Americans are divided by several such narratives. More [[w:Gun_politics_in_the_United_States|guns make us safer]], or they cause needless [[w:Gun_violence_in_the_United_States|violence]]. [[w:Climate_change|Climate change]] is the biggest threat to our future or is simply a hoax. [[w:Abortion_in_the_United_States|Abortion]] murders babies or protects a women’s constitutional right to choose. God created man, or [[w:Problem_of_the_creator_of_God|man created God]]. [[w:Psalm_115|Earth belongs to man]] or [[q:Chief_Seattle|man belongs to earth]]. Capitalism is the solution or [[w:Criticism_of_capitalism|capitalism is the problem]]. Do you believe the experts or do you [[w:Knowing_How_You_Know/Divided_by_epistemology|believe your friends]]? Various conspiracy theories provide especially troublesome narratives. Ideologies amplify [[w:List_of_cognitive_biases|cognitive biases]].
=== Powerful False Narratives ===
Very often, [[w:Storytelling|the best story wins]]. Here is an example of a powerful, influential, and harmful false narrative, known as the ''satanic panic''.

The [[w:Satanic_panic|Satanic panic]] is a [[w:Moral_panic|moral panic]] consisting of over 12,000 unsubstantiated cases of Satanic ritual abuse (SRA) starting in the United States in the 1980s, spreading throughout many parts of the world by the late 1990s, and persisting today. The panic originated in 1980 with the publication of [[w:Michelle_Remembers|''Michelle Remembers'']], a bestselling book co-written by Canadian psychiatrist [[w:Lawrence_Pazder|Lawrence Pazder]] and his patient (and future wife), Michelle Smith, which used the discredited practice of [[w:Recovered-memory_therapy|recovered-memory therapy]] to make sweeping lurid claims about satanic ritual abuse involving Smith. The allegations which afterwards arose throughout much of the United States involved reports of [[w:Physical_abuse|physical]] and [[w:Sexual_abuse|sexual abuse]] of people in the context of [[w:Occult|occult]] or [[w:Theistic_Satanism|Satanic]] rituals. In its most extreme form, allegations involve a conspiracy of a global Satanic cult that includes the wealthy and powerful world elite in which children are abducted or bred for [[w:Human_sacrifice|human sacrifices]], [[w:Child_pornography|pornography]], and [[w:Prostitution|prostitution]].

The key elements of the narrative are:
*Satanic ritual abuse is horrific and widespread.
*Unknown to us many of our children are being subjected to the horrors of satanic ritual abuse.
*The abuse is so horrible that the children [[w:Repressed_memory|repress their memories]] of the abuse and are unable to disclose their experiences.
*A newly developed interviewing technique, called [[w:recovered-memory_therapy|recovered-memory therapy]], can elicit accurate memories and testimony from the children.
*Using this technique, many children are beginning to reveal and describe the abuse they have suffered.
*This must be urgently investigated, and the abuse stopped at all costs.
*A global Satanic cult may be responsible.
*Missing memories among the victims and absence of evidence was cited as evidence of the power and effectiveness of the cult in furthering their agenda.

Initial interest arose via the publicity campaign for Pazder's 1980 book [[w:Michelle_Remembers|''Michelle Remembers'']], and it was sustained and popularized throughout the decade by coverage of the [[w:McMartin_preschool_trial|McMartin preschool trial]] and the contemporaneous [[w:Day-care_sex-abuse_hysteria|day-care sex-abuse hysteria]]. Testimonials, symptom lists, rumors, and techniques to investigate or uncover memories of SRA were disseminated through professional, popular, and religious conferences, as well as through [[w:Talk_show|talk shows]], sustaining and further spreading the moral panic throughout the United States and beyond. In some cases, allegations resulted in criminal trials with varying results; after seven years in court, the McMartin trial resulted in no convictions for any of the accused, while other cases resulted in lengthy sentences, some of which were later reversed. Scholarly interest in the topic slowly built, eventually resulting in the conclusion that the phenomenon was a moral panic, which, as one researcher put it in 2017, &quot;involved hundreds of accusations that devil-worshipping pedophiles were operating America's white middle-class suburban daycare centers.&quot; 

Of the more than 12,000 documented accusations nationwide, investigating police were not able to substantiate any allegations of organized cult abuse.

Today the [[w:Radical_right_(United_States)|far-right]] conspiracy theory movement known as [[w:QAnon|QAnon]], has adopted many of the tropes of Satanic Ritual Abuse and Satanic Panic. Instead of daycare centers being the center of abuse, however, liberal [[w:Hollywood|Hollywood]] actors, [[w:Democratic_Party_(United_States)|Democratic]] politicians, and high-ranking government officials are portrayed as a child-abusing cabal of Satanists.
=== Assignment ===
'''Part 1:'''

#Identify a powerful false narrative to study for this assignment. Choose one from this list of [[Finding Common Ground/Powerful False Narratives|powerful false narratives]], or from any other source.
#Identify the elements that make this narrative compelling, convincing, memorable, and likely to spread and be shared with others.
#Identify the falsehoods in the narrative.
#Identify any [[Embracing Ambiguity|ambiguities]] that are prematurely resolved. 
#Identify the various calls to action inspired by the narrative.
#How is this narrative harmful, if at all? 
#Who gains and who loses as this narrative spreads?

'''Part 2:'''

#Notice as conflicts arise in conversations, debates, and arguments.
#Determine if the disagreement is at the level of narrative, language, perception, or reality.
#If the narrative is driving the conversation, go through the steps above to analyze the narrative elements.
#Practice dialogue rather than debate or argumentation. Be [[w:Candor|candid]]. [[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
#Lead the dialogue toward reality and away from abstractions. Move toward the center Move toward the center (reality as it is) of the Layers of Abstraction diagram at the beginning of this course. 
#Find common ground within our [[Facing_Facts/Reality_is_our_common_ground|shared reality]].
##Don’t argue [[Facing_Facts|matters of fact]], [[Evaluating Evidence|research them]].
##[[Seeking True Beliefs|Seek true beliefs]].
##[[Embracing Ambiguity|Embrace ambiguity]].
##[[Exploring_Worldviews/Aligning_worldviews|Align your worldviews]] with reality.
##[[Transcending Conflict|Transcend conflict]].

== Ideology ==
[[File:Ideology Occludes Reality.jpg|thumb|Ideology Occludes Reality]] 
Do you think for yourself and choose your own beliefs? If you are like most people, you probably find it easier to adopt some pre-packaged set of beliefs that seem attractive. We can broadly characterize an [[w:Ideology|ideology]] as some agenda-driven set of beliefs.&lt;ref&gt;Many definitions of ideology are proposed. See, for example: [https://journals.sagepub.com/doi/10.1177/106591299705000412 Ideology: A definitional Analysis], John Gerring, December 1, 1997, Political Research Quarterly. &lt;/ref&gt;  Because ideologies are constructed to serve some agenda, an ideology is unlikely to accurately represent reality. 

An [[w:ideology|ideology]] is a set of beliefs intended to describe how the world works, or how some believe it should work. An ideology is a particular way of looking at the world, often codified into a [[w:doctrine|doctrine]]. Often our religious, political, and economic beliefs are drawn from an ideology. You may also follow particular lifestyle choices such as [[w:veganism|veganism]], or [[w:Environmentally_friendly|environmentalism]] based on a particular ideology. Ideologies substitute socially constructed models for brute facts. Many ideological models do not correspond well to reality. “Essentially, all models are wrong,” [[w:George_E._P._Box|George Box]] noted, “but some are useful.” Beware of substituting an ideology for a careful examination of reality.

As illustrated in the revised diagram shown here, ideologies impose a model that prohibits direct access to reality, and displaces any alternative narratives, interpretations, representations, or perceptions of reality. The ideology establishes all you need to know. It acts as a convenient substitute for reality. 

Immersion and commitment to an ideology can become a firmly held part of your identity. If you say “I am a Conservative” rather than “I often agree conservative political ideas” you are declaring the ideology as a part of your identity.&lt;ref&gt;{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant|date=February 2, 2021 |title=Think Again: The Power of Knowing What You Don't Know  |publisher=Viking |pages=320 |isbn=978-1984878106}}&lt;/ref&gt; This can make it harder to abandon. Although you choose your beliefs, [[True_Self#I_am|you are not your beliefs]]. 

Remaining bound by an ideological doctrine is a form of mental bondage. It is wise to break free from that bondage. Adopting a scout mindset—described below—can help us break free from ideologies that are holding our minds captive.

=== The Scout Mindset ===
[[File:U.S. Marine and Japan Ground Self-Defense Scout Snipers 170310-M-PQ336-010.jpg|thumb|Scouts seek to see the world as it is, not as they wish it was.]]
Author [[w:Julia_Galef|Julia Galef]] describes the ''scout mindset'' as “The motivation to see things as they are, not as you wish they were.”&lt;ref&gt; {{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}} @9 of 541&lt;/ref&gt;  In contrast to the ''scout mindset'', the ''soldier mindset'' is a motivation to attack differing points of view or defend a position in each argument we encounter. 
 
Contrasted with the soldier mindset, the scout mindset values:
*'''Seeing things as they''' are over defending territory or taking territory;
*'''Asking is it true?''' over arguing to defend a pre-determined position;
*'''Observations''' over interpretations;
*'''Investigation''' over argumentation;
*'''Gaining insight''' over winning this argument;
*'''Wonder''' over attacking;
*'''Learning''' over advancing a position;
*'''Observing''' over taking ground;
*'''Understanding the issue''' over shooting down arguments;
*'''Exploring possibilities''' over reinforcing a position;
*'''[[Seeking True Beliefs|Seeking true beliefs]]''' over securing long held beliefs;
*'''Seeking reality''' over defending an ideology;
*'''[[Evaluating Evidence|Objective evidence]]''' over motivated reasoning;
*'''Representative evidence''' over narratives, specious interpretations and representations;
*'''Reason''' over rhetoric;
*'''Exploration''' over staying on the ideological course;
*'''Dialogue''' over reciting [[w:Dogma|dogma]];
*'''Listening''' over reiterating and continuing to advocate;
*'''[[Virtues/Humility|Humility]]''' over arrogance;
*'''Curiosity''' over certainty or fear;
*'''[[Intellectual honesty]]''' over [[w:Prevarication|prevarication]], and
*'''Working with collaborators''' over fighting with opponents.

[[Coping with Ego|Ego]] encourages the soldier. [[Deductive_Logic/Clear_Thinking_curriculum|Reason]] encourages the scout.

The [[w:Stanford_Encyclopedia_of_Philosophy|Stanford Encyclopedia of Philosophy]] entry on Law and Ideology tells us, “Ideologies are ideas whose purpose is not epistemic, but political. Thus, an ideology exists to confirm a certain political viewpoint, serve the interests of certain people, or to perform a functional role in relation to social, economic, political, and legal institutions.” &lt;ref&gt;Sypnowich, Christine, &quot;[https://plato.stanford.edu/archives/sum2019/entries/law-ideology Law and Ideology]&quot;, The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.) &lt;/ref&gt;

Because ideologies exist to advance an agenda rather than to model reality, they require a soldier mindset to sustain the gap between the model described by the ideology and an accurate model based on reality. Ideologies are vulnerable to exploration by a scout mindset. Adopting a scout mindset is key to escaping ideologies.

=== Example Ideologies ===
We rely on many ideologies to simplify our thinking. These include: 
*[[w:Loyalty|Loyalty]], including loyalty to an individual person, a group, team, brand, causes, regions, nation, or any of the belief systems listed below. 
*[[w:List_of_mythologies|Myths and legends]],
*[[w:Folklore|Folklore]], [[w:Tradition|traditions]], [[w:Superstition|superstitions]], and [[w:Taboo|taboos]],
*[[w:Caste|Caste systems]], including [[w:Racism|racism]], [[w:Sexism|sexism]], other forms of [[w:Ascribed_status|ascribed status]], [[w:Social_class|social classes]], and other ranking constructs, 
*[[w:Stereotype|Stereotypes]], 
*[[w:Economic_ideology|Economic ideologies]], including [[w:Neoliberalism|neoliberalism]], [[w:Monetarism|monetarism]], [[w:Mercantilism|mercantilism]], [[w:Mixed_economy|mixed economy]], [[w:Social_Darwinism|social Darwinism]], [[w:Communism|communism]], [[w:Laissez-faire|laissez-faire economics]], and [[w:Free_trade|free trade]]. There are also current theories of [[w:Safe_trade|safe trade]] and [[w:Fair_trade|fair trade]] that can be understood as ideologies.
*[[w:List_of_political_ideologies|Political ideologies]], including anarchism, authoritarianism, communism, conservatism, democracy, environmentalism, fascism, separatist movements, liberalism, libertarianism, nationalism, populism, social democracy, socialism, and others.
*[[w:List_of_creation_myths|Creation myths]],
*[[w:List_of_religions_and_spiritual_traditions|Religions and spiritual traditions]], 
*[[w:Quackery|Quackery]],
*[[w:List_of_topics_characterized_as_pseudoscience|Pseudoscientific beliefs]],
*[[w:Paranormal|Paranormal beliefs]],
*Commitment to [[w:List_of_conspiracy_theories|conspiracy theories]], and
*[[w:List_of_new_religious_movements|New religious movements]].
=== Assignment ===
'''Part 1:'''

#Identify each of the ideologies you identify with, belong to, or agree with. Use the list above as a guide or use any other method to identify ideologies that influence you.
#For each of the ideologies identified in step 1:
##Decide if the model it presents is an accurate representation of reality in all its scope and complexity.
##Identify any [[Embracing Ambiguity|ambiguities]] that are prematurely resolved. 
##Decide if the ideology is helping you understand reality, or is occluding, limiting, biasing, or censoring your understanding of reality.
##If you decide a particular ideology is unhelpful, take steps to abandon that ideology. 

Welcome those who are [[Seeking True Beliefs|seeking truth]]. Abandon those who are [[w:Certainty|certain]] they have found [[w:Dogma|Truth]].

'''Part 2:'''
#Study the module on [[Knowing_How_You_Know/Examining_Ideologies|Examining Ideologies]] within the [[Knowing How You Know|Knowing how you know]] course.
#Complete the [[Knowing_How_You_Know/Examining_Ideologies#Assignment|assignments]] in that module.
#Read the essay [[/Every Ism Creates a Schism/|&quot;Every Ism Creates a Schism&quot;: An Exploration]].
#Think beyond the doctrine.

== Toward Ought ==
[[File:Compass rose browns 00.png|thumb|right| 250px|A deep understanding of [[Moral_Reasoning#A_Basis_for_Moral_Reasoning |''impartiality'']] can guide toward what we ought to do. ]]
So far in this course, the common ground we have considered is our shared reality. This is our collective understanding of ''what is'' in the world. Can we find a corresponding common ground regarding what we ''ought to do''?

Philosopher [[w:David_Hume|David Hume]] famously observed that knowing only ''what is'', we cannot determine what we [[w:Is–ought problem|''ought to do'']]. However, by making the reasonable [[Moral_Reasoning#A_Basis_for_Moral_Reasoning |assumption of ''impartiality'']], we can begin to identify what it is we ought to do. 

We ought to [[Assessing_Human_Rights/Beyond_Olympic_Gold|advance human rights worldwide]], adopt [[Level_5_Research_Center#Values|pro-social values]], and establish a well-founded basis for [[Moral Reasoning|moral reasoning]].

=== Assignment ===
#Complete the Wikiversity course on [[Assessing Human Rights]].
#[[Assessing_Human_Rights/Beyond_Olympic_Gold|Advance human rights, worldwide]]. 
#Study this [[Level_5_Research_Center#Values|list of pro-social values]].
#Adopt pro-social values and [[Level_5_Research_Center/Choosing_Level_5_Living|choose level 5 living]]. 
#Complete the Wikiversity course on [[Moral Reasoning]].
#Develop your own well-founded basis for [[Moral Reasoning|moral reasoning]].

== Simple but not easy ==
The central idea in this course—[[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]]—is simply stated and supported by overwhelming evidence. However, it may not be easy for you, or others you encounter or care about to change deeply held beliefs that differ from objective reality. Here are some steps you can take to align your beliefs with reality and work to find common ground with others.
#Begin by aligning your own beliefs with reality. Complete the course on [[Seeking True Beliefs]]. [[Exploring_Worldviews/Aligning_worldviews|Align your own worldview]] with reality. [[Deductive_Logic/Clear_Thinking_curriculum|Think clearly]]. [[Knowing How You Know|Know how you know]]. Become skillful at [[Evaluating Evidence|evaluating evidence]]. Stay curious. [[Embracing Ambiguity|Embrace ambiguity]].
#Be careful to [[Facing_Facts#Degrees_of_Consensus|distinguish among matters of fact]], matters of controversy, and matters of opinion. Do not argue matters of fact, research them. Do not argue matters opinion, enjoy them. Reason carefully and listen closely when discussing matters of controversy. Although [[Virtues/Tolerance|tolerance]] is essential in matters of opinion, it has no place in [[Facing Facts|matters of fact]].
#Spend the required effort to prepare to find common ground with others. 
##Complete the course on [[Practicing Dialogue|practicing dialogue]]. Practice dialogue.
##Ensure all participants have adopted a [[Socratic_Methods#Essential_Socratic_Temperament|Socratic temperament]].
##Expect [[intellectual honesty]].
###Complete the course on [[intellectual honesty]].
###[[Living Wisely/Advance no falsehoods|Advance no falsehoods]].
###Discuss the importance of [[intellectual honesty]]—accurately describing your true beliefs—with each participant. Gain agreement to be intellectually honest, and to expect intellectual honesty from all participants. Do not indulge [[w:Charlatan|charlatans]]. Do not [[w:Internet_troll|feed the trolls]]. 
#Finding common ground often challenges deeply held beliefs. This is uncomfortable; people often react with fear and may seek to withdraw from the dialogue or divert the conversation.  Notice the shift from [[Finding_Common_Ground#The_Scout_Mindset|scout mindset]] to soldier mindset. Notice when fear begins to displace curiosity. Encourage the fearful person to allow curiosity to displace fear. Return to a scout mindset.
#Real good is our common ground. People who are [[Living Wisely/Seeking Real Good|seeking real good]] will find common ground with others who are also seeking real good. When [[Transcending Conflict|conflict arises]], it is likely because someone is not seeking real good. 
#The course on [[Street Epistemology]] provides specific techniques for exploring the basis for beliefs. Use applicable Street Epistemology techniques when seeking common ground.

== Summary and Conclusions ==
Reality exists. Reality is vast, complex, and dynamic. Humans have only investigated a small portion of the universe, and our investigation is incomplete. We all live together on this one planet we call Earth and [[Knowing_How_You_Know/One_World|we all live in the same universe]].

Because we all live in the same universe, our reliable understanding of that universe must eventually converge toward one coherent description. Because reality exists, we can [[Evaluating_Evidence|examine reality]], and we can [[Exploring_Worldviews/Aligning_worldviews|align our worldview]] with reality. Building on these observations, it follows that [[Facing_Facts/Reality_is_our_common_ground|reality is our common ground]], even though any number of worldviews is possible.

[[w:Perception|Perception]] transforms sensory information originating with some material object, known as the ''target'' or the ''[[w:Distal|distal]] stimulus'' into mental representations known as [[w:Perception#Process_and_terminology|percepts]], which do not exist elsewhere in the world. 

[[w:Perception|Perception]]—the process of extracting information from energy that impinges on our [[w:Sense|sensory organs]]—is not straightforward. There is more to perception than meets the eye.

Our most direct encounters with reality are through direct sensory observations and awareness. However, our perceptions are not exact replicas of the distal objects being perceived. Omissions, distortions, and additions take place. Fortunately, we can augment our direct perceptions to obtain a more accurate understanding of our world. Furthermore, [[Facing_Facts/Perceptions_are_Personal|perceptions are personal]], and it is often a [[w:Problem_of_induction|mistake to generalize]] our personal perceptions beyond our own experiences.

We represent our thoughts using various symbols, including words, gestures, facial expressions, images, and analogies. The mapping from various representations to reality is imprecise. Language and other symbols can be constraining, ambiguous, refer to social constructs, prejudiced, based on artificial boundaries, or focus on convenient representation rather than reality. The map is not the territory, although we often confuse our representations for realty.

People [[w:Interpretation_(philosophy)|interpret]] observations to resolve ambiguity, increase certainty, and to account for observations within some familiar model or analogy. The role of interpretation is most evident when the items being interpreted are most [[w:Ambiguity|ambiguous]]. 

[[w:Tasseography|Tasseography]]—reading tea leaves—reveals the essence of interpretation. Because the tea leaf formations are entirely arbitrary the interpreter can say almost anything. The interpretation reflects judgements, evaluations, biases, concepts, models, analogies, predictions, optimism, pessimism, certainty, risk, doubt, hope, fears, storytelling skills, and desires of the interpreter, entirely independent of the tea leaf formation.

The [[w:Rabbit–duck illusion|rabbit-duck illusion]] allows for two different, yet equally correct interpretations. Life is often more complicated than ducks and rabbits, and we can reasonably differ in interpreting many words.

Interpretations that prematurely eliminate [[w:Doubt|doubt]] and resolve [[w:Ambiguity|ambiguity]] into a comfortable feeling of [[w:Certainty|certainty]] have led to several tragedies. Become [[Embracing Ambiguity|comfortable with ambiguity]].

Humans enjoy telling and retelling [[w:Narrative|stories]]. This may be the defining characteristic of the human species. We are exposed to many powerful false narratives. To find common ground we must dismiss the falsehoods in narratives.

An [[w:ideology|ideology]] is a set of beliefs intended to describe how the world works, or how some believe it should work. An ideology is a particular way of looking at the world, often codified into a [[w:doctrine|doctrine]]. Often our religious, political, and economic beliefs are drawn from an ideology.

Ideologies substitute socially constructed models for brute facts. Many ideological models do not correspond well to reality. “Essentially, all models are wrong,” [[w:George_E._P._Box|George Box]] noted, “but some are useful.” Beware of substituting an ideology for a careful examination of reality.

Remaining bound by an ideological doctrine is a form of mental ''bondage''. It is wise to break free from that bondage. Adopting a ''scout mindset'' can help us break free from ideologies that are holding our minds captive.

It is likely your beliefs are influenced by [[Knowing How You Know/Examining Ideologies|inaccurate ideologies]]. Identify these and abandon them.

This is not about compromise. This is about gaining an accurate understanding of the world we live in.

[[Facing Facts/Reality is the Ultimate Reference Standard|Reality is our reference standard]]. 

Embrace reality. Seek true beliefs. Dismiss misleading perceptions, representations, interpretations, narrations, and ideologies. Align concepts with reality.

[[Assessing_Human_Rights/Beyond_Olympic_Gold|Advance human rights worldwide]], adopt [[Level_5_Research_Center#Values|pro-social values]], and establish a well-founded basis for [[Moral Reasoning|moral reasoning]].

[[Transcending_Conflict|Transcend conflict]].

[[Living_Wisely/Seeking_Real_Good|Seek real good]].

It is difficult to change deeply held beliefs, however if ''you'' can do this, ''they'' can do this.

We can [[Facing_Facts/Reality_is_our_common_ground|find common ground]].

== Recommended Reading ==
*{{Cite book|title=Truth: what it is, how to find it, and why it still matters|publisher=Johns Hopkins University Press|date=2026|location=Baltimore|isbn=978-1-4214-5372-9|first=Michael|last=Shermer}}
*{{cite book |last=Van der Stigchel |first=Stefan |date=March 12, 2019 |title=How Attention Works: Finding Your Way in a World Full of Distraction |publisher=The MIT Press |pages=152 |isbn=978-0262039260 }}
*{{cite book |last=Rogers |first=Brian |date=December 26, 2017 |title=Perception: A Very Short Introduction |publisher=Oxford University Press |pages=144 |isbn=978-0198791003 }} 
*{{cite book |last1=Gause  |first1=Donald C. |last2=Weinberg |first2=Gerald M. |date=March 1, 1990 |title=Are Your Lights On?: How to Figure Out What the Problem Really Is  |publisher=Dorset House Publishing Company |pages=176 |isbn=978-0932633163 |author-link=w:Gerald_Weinberg}} 
*{{cite book |last=Weinberg |first=Gerald M. |author-link=w:Gerald_Weinberg |date=September 1, 1989 |title=Exploring Requirements: Quality Before Design  |publisher=Dorset House Publishing Company |pages= 320 |isbn=978-0932633132}}
*{{cite book |last=Holmes |first=Jamie |author-link=w:Jamie_Holmes_(author) |date=October 11, 2016 |title=Nonsense: The Power of Not Knowing Paperback  |publisher=Crown |pages=336 |isbn=978-0385348393}}
*{{cite book |last=Ariely |first=Dan |author-link=w:Dan_Ariely |date=September 17, 2024 |title=Misbelief: What Makes Rational People Believe Irrational Things |publisher=Harper Perennial  |pages=320 |isbn=978-0063280434}}
*{{cite book |last=Burton M.D. |first=Robert A.  |date=Mar 17, 2009 |title=On Being Certain: Believing You Are Right Even When You're Not |publisher=St. Martin's Griffin |pages272 |isbn=978-0312541521}}
*{{cite book |last=Duke |first=Annie |author-link=w:Annie_Duke |date= |title=Thinking in Bets: Making Smarter Decisions When You Don't Have All the Facts |publisher=Portfolio |pages= 288 |isbn=978-0735216372}}
*{{cite book |last=Freinacht  |first=Hanzi  |date=March 10, 2017 |title=The Listening Society: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=414 |isbn=978-8799973903}}
*{{cite book |last=Freinacht  |first=Hanzi  |date=May 29, 2019 |title=Nordic Ideology: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=495 |isbn=978-8799973927}}
*{{cite book |last1=Gilovich |first1=Thomas |last2=Ross |first2=Lee |date=December 1, 2015 |title=The Wisest One in the Room: How You Can Benefit from Social Psychology's Most Powerful Insights|publisher=Free Press|pages=320|isbn=978-1451677546}} 
*{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}}
*{{cite book |last=Hawkins |first=Jeff  |author-link=w:Jeff_Hawkins |date=March 2, 2021 |title=A Thousand Brains: A New Theory of Intelligence |publisher=Basic Books |pages=288 |isbn=978-1541675810}}
*{{cite book |last1=Hofstadter |first1=Douglas R  |last2=Sander |first2=Emmanuel |author-link=w:Douglas_Hofstadter |date=April 23, 2013 |title=Surfaces and Essences: Analogy as the Fuel and Fire of Thinking  |publisher=Basic Books |pages=592 |isbn=978-0465018475}}
*{{cite book |last= Weinberg |first=Gabriel  |date=June 18, 2019 |title=Super Thinking: The Big Book of Mental Models |publisher=Portfolio |pages=352  |isbn=978-0525533580}}
*{{cite book |title=The Art of Possibility: Transforming Professional and Personal Life |last1=Stone Zander |first1=Rosamund |last2=Zander|first2=Benjamin   |year=224 |publisher=Penguin |isbn=978-0142001103  |pages=224}}
*{{cite book |last1=Lakoff |first1=George |last2=Johnson |first2=Mark|date=April 15, 2003 |title=Metaphors We Live By |publisher= |pages=242 |isbn=978-0226468013 |author-link=w:George Lakoff }}
*{{cite book |last1=Heath |first1=Chip |last2=Heath |first2=Dan  |author-link=w:Chip_Heath|date= |title=Made to Stick: Why Some Ideas Survive and Others Die   |publisher=Random House|pages= 291 |isbn=978-1400064281}} 
*{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant|date=February 2, 2021 |title=Think Again: The Power of Knowing What You Don't Know  |publisher=Viking |pages=320 |isbn=978-1984878106}} 
*{{cite book |last1=Campbell |first1=Joseph | last2=Moyers |first2=Bill |date=June 1, 1991 |title=The Power of Myth  |publisher=Anchor |pages=293 |isbn=978-0385418867 |author-link=w:Joseph_Campbell }}
*{{cite book |last=Mackay |first=Charles |date=November 1, 2016 |title=[[w:Extraordinary Popular Delusions and The Madness of Crowds|Extraordinary Popular Delusions and The Madness of Crowds]] |publisher=CreateSpace Independent Publishing Platform |pages=386 |isbn=978-1539849582 |author-link=w:Charles_Mackay_(author) }}
*{{cite book |last=Orwell |first=George  |date=March 1, 2017 |title=[[w:Animal Farm|Animal Farm]] |publisher=Fingerprint! Publishing |pages=152 |isbn=978-9386538284 |author-link=w:George_Orwell }}
*{{cite book |last=Orwell  |first=George |date=January 15, 2013 |title=[[w:Politics and the English Language|Politics and the English Language]] |publisher=Penguin Classic |pages=48 |isbn=978-0141393063 |author-link=w:George_Orwell }}
*{{cite book |last=Hecht |first=Jennifer Michael |author-link=w:Jennifer_Michael_Hecht |date=September 7, 2004 |title=Doubt: A History: The Great Doubters and Their Legacy of Innovation from Socrates and Jesus to Thomas Jefferson and Emily Dickinson |publisher=HarperOne |pages=576 |isbn=978-0060097950}}
*{{cite book |last=Carroll |first=Sean M |author-link=w:Sean_M._Carroll |date=May 4, 2017 |title=The Big Picture: On the Origins of Life, Meaning, and the Universe Itself  |publisher=Oneworld Publications |pages=480 |isbn=978-1786071033}}
*{{cite book |last=Pinker |first=Steven |author-link=w:Steven_Pinker |date=February 13, 2018 |title=Enlightenment Now: The Case for Reason, Science, Humanism, and Progress |publisher=Viking |pages=576  |isbn=978-0525427575}}
*{{cite book |last=Wilczek |first=Frank |author-link=w:Frank_Wilczek |date=January 12, 2021 |title=Fundamentals: Ten Keys to Reality  |publisher=Penguin Press |pages=272 |isbn=978-0735223790}}
*{{cite book |last=Gray |first=Dave |author-link= |date=September 14, 2016 |title=Liminal Thinking: Create the Change You Want by Changing the Way You Think  |publisher=Two Waves Books |pages=184 |isbn=978-1933820460}}
*{{cite book |last=Schulz |first=Kathryn |author-link=w:Kathryn_Schulz |date=June 8, 2010 |title=Being Wrong: Adventures in the Margin of Error |publisher=Ecco |pages=416 |isbn=0061176044}}
*{{cite book |last=Temple |first=David J.  |date=April 2, 2024 |title=First Principles and First Values: Forty-Two Propositions on Cosmoerotic Humanism, the Meta-Crisis, and the World to Come  |publisher=World Philosophy &amp; Religion Press |pages=296 |isbn=979-8989588909}}

I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research.

*{{cite book |last=Coleman |first=Peter T. |author-link=w:Peter_T._Coleman_(academic) |date=June 1, 2021 |title=The Way Out: How to Overcome Toxic Polarization  |publisher=Columbia University Press |pages=296 |isbn=978-0231197403}}

== References ==
&lt;references/&gt;

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[[Category:Reality]]
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    <title>Responding to a nuclear attack</title>
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      <comment>improve description esp of plot</comment>
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      <text bytes="36950" sha1="jzor62kjqjen4qbhmi164iiqdyqeror" xml:space="preserve">{{Research project}}

:''This brief note is on Wikiversity to invite others to provide alternative responses to this question, adding relevant, substantive references, moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- and raising other questions and concerns on the associated [[Wikiversity:FAQ|''''“Discuss”'''' page]].'' 
::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' 

What's the best response to a nuclear attack?  

That's a difficult question.  The opposite is much easier:  

* '''''What's the ''worst'' response to a nuclear attack?'''''  

[[File:How would a nuclear war between Russia and the US affect you personally? - Future of Life Institute.webm|thumb|Simulation of a nuclear war between Russia and the US.&lt;ref&gt;Tegmark (2023).&lt;/ref&gt;]]
::The evidence summarized in this article suggests that the ''worst'' worst response to a nuclear attack would be '''a nuclear response''', because it would increase the death toll from millions to billions, the vast majority of whom would be civilians, and many and likely most of those would be in countries not directly involved in the nuclear exchange. 
::If you think otherwise, please revise this article accordingly, subject to the standard Wikimedia Foundation rules of writing from a neutral point of view citing credible sources.  Or post your concerns to the &quot;Discuss&quot; page associated with this article.    

[[File:Percent of the world's population dead from a nuclear war.svg|thumb|Percent of the world's population dead from a nuclear war on the vertical axis vs. total megatonage of nuclear weapons detonated ranging from 0.5 to 440 on the bottom axis and teragrams (millions of metric tons) of soot lofted to the stratosphere ranging from 5 to 150 on the top axis. These are from simulations by an international team of 10 scientists who specialize in modeling climate, food production, and economics&lt;ref&gt;Xia et al. (2022; see esp. their Table 1).&lt;/ref&gt; with models fit thereto. The direct deaths range between 5 and 10 percent of the total, most of who would starve to death within two years of the nuclear war.&lt;ref&gt;Xia et al. (2022, Table 1) reported &quot;Number of direct fatalities&quot; and &quot;Number of people without food at the end of year 2&quot; out of a total population of 6.7 billion for their simulated year 2010. Xia et al. (2022, Fig. 1) show that the climate impact does not start recovering until year 5 after the nuclear war and has not yet fully recovered 9 years after the war.  Thus, few people still alive without food at the end of year 2 will not likely live to year 9.  Second, the percentages plotted here are the sums of those two numbers divided by 6.7 billion.  The Wikipedia article on [[w:World population|World population]] said the world population in 2010 was 6,985,603,105 -- 7 billion (accessed 2023.08-12).  The difference between 6.7 and 7 billion seems so slight that it can be safely ignored, especially given the uncertainty inherent in these simulations and the likelihood that the small populations excluded were probably not substantively different from those included.&lt;/ref&gt; &quot;IND-PAK&quot; marks a range of hypothetical nuclear wars between [[w:India and weapons of mass destruction|India]] (IND) and [[w:Pakistan and weapons of mass destruction|Pakistan]] (PAK).  &quot;USA-RUS&quot; marks a simulated nuclear war between [[w:Nuclear weapons of the United States|the US]] (USA) and [[w:Russia and weapons of mass destruction|Russia]] (RUS).  &quot;PRK&quot; = a simulated nuclear war in which [[w:North Korea and weapons of mass destruction|North Korea]] (the People's Republic of Korea, PRK) used 30 nuclear weapons with an average yield of 17 kt for a total of 510 kt (0.51 megatons), the lower end of the bottom scale, with no nuclear retaliation by an adversary, as recommended in this article.&lt;ref&gt;Estimates of North Korea's nuclear very widely. The wikipedia article on &quot;[[w:North Korea and weapons of mass destruction|North Korea and weapons of mass destruction]]&quot; said they had 60 nuclear weapons when accessed 2026-05-20 but only half that when accessed 2023-08-07.&lt;/ref&gt;]]

This conclusion is supported by the accompanying plot summarizing climate simulations by an international interdisciplinary team of 10 scientists who specialize in mathematical and statistical modeling of climate, food production, and economics.  Five of their scenarios describe hypothetical nuclear wars between India and Pakistan that loft between 5 and 47 Tg (teragrams = millions of metric tons) of smoke (soot) to the stratosphere, where it will linger for years covering the globe and reducing the amount of solar radiation reaching the earth.  That in turn will substantially reduce the production of food for humans.  The resulting impact on the global economy means that between 4 and 40 percent of humanity will likely starve to death if they do not die of something else sooner. A hypothetical nuclear war between the US and Russia could lead to the deaths of between 80 and 85 percent of humanity with death tolls of roughly 99 percent in the US, Russia, Europe, and China.  In any of these scenarios, between 5 and 10 percent of the fatalities would result from bomb blasts. The remaining 90 to 95 percent starved to death (and presumably other sources like radiation poisoning and increased risks of disease).&lt;ref&gt;Xia et al. (2022, esp. their Tables 1 and 2). Their Table 1 gives numbers of fatalities out of a total 2010 &quot;population of the nations used in this study [of] 6,700,000,000.&quot; They give 2 simulations of a nuclear war between the US and Russia. Both would produce an estimated 360 million direct fatalities and loft 150 Tg (teragrams = million metric tonnes) to the stratosphere. At the end of the second year after such a war, between 5.08 and 5.34 billion people would be without food, totaling between 5.44 and 5.70 billion presumed dead. Those numbers are 81 and 85 percent of the 6.7 billion in the study and 78 and 81 percent of the 2010 [[w:World population|world population]] of 7 billion. We assume that the impact on the 300 million humans not in this study will not be substantively different from the 6.7 billion included and therefor use the 80-85 percent figures.&lt;/ref&gt;

This claim is clearer, more succinct, and stronger than the [[Wikisource:Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races|Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races]], &quot;that a nuclear war cannot be won and must never be fought&quot;, issued 2022-01-03 by the leaders of the first five nuclear-weapon states.&lt;ref&gt;[[Wikisource:Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races]].  See also Borger (2022). Douthat (2022) discussed the [[w:2021-2022 Russo-Ukrainian crisis|current Ukraine crisis]] in [[w:The New York Times|''The New York Times'']].  He concluded that for us (presumably the US and perhaps its NATO allies) &quot;To escalate now against a weaker adversary [Russia], one less likely to ultimately defeat us and more likely to engage in atomic recklessness if cornered, would be a grave and existential folly.&quot;&lt;/ref&gt; This repeated a statement made 1987-12-11 by US President [[w:Ronald Reagan| Ronald Reagan]] and USSR head of state [[w:Mikhail Gorbachev|Mikhail Gorbachev]].&lt;ref&gt;&lt;!-- Joint statement by Reagan, Gorbachev --&gt;{{cite Q|Q111845607}}  Reagan made that same statement 1984-01-25 in his [[Wikisource:Ronald Reagan's Fourth State of the Union Address|fourth State of the Union Address]].&lt;/ref&gt; 

In the following we review the evidence for and against this claim and then comment on the credibility of the logic that led to the creation of the world's current nuclear arsenals and seems to be driving the current &quot;modernization&quot; programs in the US, Russia, China and elsewhere.  

== Summary of research on the consequences of a nuclear war ==

It is theoretically possible that a nuclear exchange would end like [[w:World War II|World War II]] with no more than [[w:Atomic bombings of Hiroshima and Nagasaki|two nuclear weapons being used]].  It is also theoretically possible that nuclear weapons in a new war would only target deserted areas like [[w:List of nuclear weapons tests|the locations where more than 2,000 tests of nuclear weapons]] have been conducted so far.&lt;ref&gt;For a &quot;[[w:List of nuclear weapons tests|List of nuclear weapons tests]]&quot;, see the Wikipedia article by that title (accessed 2023-07-06).&lt;/ref&gt; Either of those scenarios would increase the level of harmful background radiation worldwide leading to increases in the rates of cancer, birth defects and genetic mutations, but would otherwise not likely have an immediate impact on a large portion of humanity.&lt;ref&gt;Johnston (2001) reported that only 521 of the more than 2,000 nuclear weapons tests were above ground.  If 521 explosions of nuclear weapons in deserted places have not generated a substantive impact on human health, it seems unlikely that a nuclear war involving a few thousand explosions of nuclear weapons in deserted areas would be dramatically worse.&lt;/ref&gt;
  
However, a nuclear war with such negligible results is highly unlikely.  More likely is the deaths in a few hours or days of tens or hundreds of millions of humans.&lt;ref&gt;The &quot;Number of direct fatalities&quot; in a nuclear war lasting a week ranged from 27 to 360 million in simulations summarized in Xia et al. (2022, Table 1).&lt;/ref&gt; More would die of radiation poisoning over the next few months and years.&lt;ref&gt;Ellsberg (2017, pp. 2-3) includes a graph that the Joint Chiefs Joint Chiefs of Staff produced in the Spring of 1961 to answer President Kennedy's question, &quot;If your plans for a general [nuclear] war are carried out as planned, how many people will be killed in the Soviet Union and China?&quot; This graph was a straight line beginning at 275 million who would die during the initial nuclear exchange with another 8.25 million dying each month for the next six months, totaling 325 million deaths.&lt;/ref&gt; If more than a few dozen nuclear weapons are used, then &quot;nuclear war would also produce nearly instantaneous climate change that among other effects, would threaten the global food supply.  Even a regional nuclear war ..., such as between India and Pakistan,&lt;ref&gt;Robock et al. (2007); Toon et al. (2019).  Of course, a nuclear war could be started accidentally by any nuclear-weapons state, as suggested in the report of an Indian cruise missile that landed 2022-03-10 in Pakistan (Mashal and Masood 2022). See also Xia et al. (2022).&lt;/ref&gt; in which less than 3% of the world’s nuclear weapons stockpiles were detonated in urban areas, would suddenly decrease the average global temperature by 1°C–7°C [2°–13°F], precipitation by up to 40%, and sunlight by up to 30%. ... Such a conflict would decrease crop production to an extent that it could seriously threaten world food security and even trigger global famine&quot;,&lt;ref&gt;Jägermeyr et al. (2020).&lt;/ref&gt; according to Robock and Prager (2021).  In theory, crop losses of between 10 and 25 percent for 5-10 years&lt;ref&gt;as predicted by Jägermeyr et al. (2020) and others.&lt;/ref&gt; might not threaten a global famine or even an increase in malnutrition if people ate more plant-based foods and less meat.  In practice, famines never work that way:  There is hoarding, and many who do not die of starvation succumb to diseases or secondary wars driven by the food insecurity, according to Helfand (2013).  [[w:Amartya Sen|Nobel Prize Economist Sen]] observed that, &quot;no famine has ever taken place ... in a functioning democracy&quot;.&lt;ref&gt;Sen (1999, p. 32). Later on p. 178, he stated similarly, &quot;there has never been a famine in a functioning multiparty democracy.&quot;&lt;/ref&gt; This generalizes the observation that Ireland was a ''net food exporter'' during its infamous potato famines of the nineteenth century.&lt;ref&gt;e.g., Woodham-Smith (1962).&lt;/ref&gt; Xia et al. (2022, Table 1) estimated that between 4 and 85 percent of humanity would starve to death if they did not die of something else sooner in the nuclear wars they simulated. 

In the spring of 1961, &quot;The total death toll as calculated by the Joint Chiefs of Staff [top US military leaders], from a U.S. first strike aimed at the Soviet Union, its Warsaw Pact satellites, and China, would be roughly six hundred million dead.  A hundred Holocausts&quot;, according to Daniel Ellsberg, who served as a nuclear war planner for presidents Eisenhower, Kennedy, Johnson and Nixon&lt;ref&gt;Ellsberg (2017, esp. pp. 2-3) noted that 325 million would die in the Soviet Union and China and another couple hundred million in neighboring countries, totalling six hundred million.&lt;/ref&gt; before releasing [[w:The Pentagon Papers|&quot;The Pentagon Papers&quot;]] in 1971.  Six hundred million was roughly 20 percent of the total human population on earth in 1961, and that didn't count any in the US who might be killed in retaliation.  In 1957, roughly 4 years earlier, Mao Zedong, then the Chairman of the People's Republic of China, had reportedly said that a nuclear war could kill a third of humanity, perhaps half, &quot;but imperialism would be razed to the ground, and the whole world would become socialist.&quot;&lt;ref&gt;Dikötter (2010).  See also Halimi (2018), which gives the date as 1957.  There is some controversy about this quote;  see the Wikipedia article on [[w:Mao Zedong|&quot;Mao Zedong&quot;]], accessed 2022-03-02.&lt;/ref&gt;

Turco et al. (1983) published the first predictions of a ''[[w:nuclear winter|nuclear winter]]'' based on climate modeling that considered smoke anticipated from fires started by a massive nuclear weapons exchange between the US and the Soviet Union. They found that &quot;average light levels can be reduced to a few percent of ambient and land temperatures can reach -15° to -25°C [5° to -4°F]&quot; with smoke transported from the Northern to the Southern Hemisphere, all of which &quot;could pose a serious threat to human survivors and to other species.&quot;  Various teams have published comparable analyses since then with different and increasingly sophisticated models, beginning with Aleksandrov and Stenchikov (1983), with similar conclusions.&lt;ref&gt;Coup et al. (2019, p. 8522).&lt;/ref&gt; Coup et al. (2019) predicted hard freezes ''in the summer'' in most of the Northern Hemisphere including the US, Russia, and most of Europe during the first three years following such a war, where temperatures drop below −4°C [25°F], making it impossible to grow crops in those regions.  China would suffer a similar fate, with only its southeast portion remaining above freezing in the summer.  Much of Southern Mexico, Central and South America, and the Southern Hemisphere would also be negatively impacted, but not to the same extent.  These climate modeling results make Mao's predictions from 1957 seem wildly optimistic:  Any humans in the US, Canada, or most of Eurasia who survived the nuclear exchange would have extreme difficulties finding enough to eat -- &quot;imperialism razed to the ground&quot;, according to Mao.  However, crop yields in most of the rest of the world would also be extremely depressed, which Mao had not considered.  The results would threaten famine vastly worse than what has been predicted following a nuclear war between India and Pakistan.&lt;ref&gt;Ellsberg said that 98 or 99 percent of humanity would starve to death if they did not die of something else sooner (Ellsberg et al. 2017).  Coup et al. (2019) and Xia et al. (2022) conclude that it won't be quite that bad but will still pretty grim.&lt;/ref&gt; 

Of course, no one knows for sure how many people would die directly and indirectly from a nuclear war.  However, it should be obvious to at least some if not most people that the ''worst'' response to a nuclear attack would be a nuclear response:  

* A nuclear response to a nuclear &quot;warning shot&quot; with minimal destruction could too easily escalate until the nuclear arsenals of all parties were expended and the life expectancy of all survivors worldwide was dramatically reduced.  

* Alternatively, a nuclear response to a massive first strike against a thousand cities would most likely ''increase'' the death toll and reduce the life expectancy of survivors ''in the country responding with nuclear weapons'' (and, of course, in other countries not officially involved).  

* It is possible that a nuclear response could deter further uses of nuclear weapons and reduce the length and severity of the war and its global impact.  However, this outcome seems unlikely given the record of history.  

Turcotte (2022) concluded that if the 2022 Ukraine 'conflict ends without the annihilation of our species, it should nonetheless be regarded as a planet-wide near-death experience, and the “Peoples of the United Nations” should demand the total elimination of nuclear weapons as quickly as humanly possible, as well as the establishment of new common security measures that will move us much closer to sustainable peace throughout the world.' In spite of this concern, Turcotte recommended military action to support Ukraine but short of declaring war on Russia.

Leading experts have made alarming comments about the likelihood of a nuclear attack, possibly by a terrorist organization.  In 2004 Bruce Blair, president of the [[w:Center for Defense Information|Center for Defense Information]] wrote: &quot;I wouldn't be at all surprised if nuclear weapons are used over the next 15 or 20 years, first and foremost by a terrorist group that gets its hands on a [[w:Russia and weapons of mass destruction|Russian]]&quot; or [[w:Pakistan and weapons of mass destruction|Pakistani nuclear weapon]].&lt;ref&gt;&lt;!--Nicholas D. Kristof (2004) A Nuclear 9/11, NYT--&gt;{{cite Q|Q111906710}}&lt;/ref&gt;

Other experts seemed even more concerned:  A nuclear terrorist attack in the US was considered &quot;more likely than not&quot; within the next five to ten years, according to Professor [[w:Robert Gallucci|Robert Gallucci]] of the [[w:Georgetown University School of Foreign Service|Georgetown University School of Foreign Service]] in 2006 or in the next decade per former U.S. Assistant Secretary of Defense [[w:Graham Allison|Graham Allison]] in 2004.&lt;ref&gt;&lt;!-- Ordre Kittrie (2007) Averting Catastrophe: Why the Nuclear Non-proliferation Treaty is Losing its Deterrence Capacity and How to Restore It --&gt;{{cite Q|Q111906652}}&lt;/ref&gt; 

The Wikipedia article on &quot;[[w:National Response Scenario Number One|National Response Scenario Number One]]&quot; describes &quot;the United States federal government's planned response to a nuclear attack.&quot;  It focuses primarily on &quot;the possible detonation of a small, crude nuclear weapon by a terrorist group in a major city, with significant loss of life and property.&quot;&lt;ref&gt;Accessed 2022-05-08, when it cited &lt;!-- Jay Davis (2008) After A Nuclear 9/11 --&gt;{{cite Q|Q111905675}}, &lt;!-- Brian Michael Jenkins (2008) A Nuclear 9/11? --&gt;{{cite Q|Q111906145}}&lt;/ref&gt; That article discusses preparing for a nuclear attack but not how to respond.  

Nevertheless, if the ''worst'' response to a nuclear attack is a nuclear response, that has other policy implications for leaders of nuclear ''and non-nuclear'' countries world wide.  However, an analysis of those implications will be left for future work.&lt;ref&gt;Turcotte (2022) offered some suggestions.  Recommendations more consistent with the analysis here is the &lt;!--Veterans For Peace Nuclear Posture Review
--&gt;{{cite Q|Q111141993}} They mention the &quot;[[w:Treaty on the Prohibition of Nuclear Weapons|Treaty on the Prohibition of Nuclear Weapons]]&quot;, supported by the [[w:International Campaign to Abolish Nuclear Weapons|International Campaign to Abolish Nuclear Weapons (ICAN)]].&lt;/ref&gt;

== Credibility of military leaders and national security experts ==

{{main|Expertise of military leaders and national security experts}}

* ''Never attribute to malice that which is adequately explained by stupidity.'' ([[w:Hanlon's razor|Hanlon's razor]])
* ''Never attribute to malice or stupidity that which can be explained by moderately rational individuals following incentives in a complex system.'' (Hubbard's clumsier correlary.&lt;ref&gt;Hubbard (2020, pp. 81-82).&lt;/ref&gt;)

The history of armed conflict should raise questions about the credibility of those advocating use of military force:  In all major armed conflicts in history, at least one side has lost.  Often the official winners lost substantially more than they gained.  

=== Research on expertise === 

The history of armed conflict is consistent with the research by Kahneman and Klein (2009) in their conclusion that 

:''expert intuition is learned from frequent, rapid, high-quality feedback.''  

In particular, military leaders in combat can get frequent, rapid high-quality feedback on their ability to deliver death and destruction to designated targets. However, no one can get such feedback about how to win wars or how to ''promote broadly shared peace and prosperity for the long term.'' This is discussed in more detail in the Wikiversity article on &quot;[[Expertise of military leaders and national security experts]]&quot;. That article documents how experts without such feedback can be beaten by simple rules of thumb developed by intelligent lay people.&lt;ref&gt;Kahneman et al. (2021) report that with some data, a statistical model fit often does better. With lots of data, artificial intelligence systems can do even better. This extends the work of [[w:Paul E. Meehl#Clinical versus statistical prediction|Meehl (1954)]]. Hubbard (2020) and [[w:Superforecasting: The Art and Science of Prediction|Tetlock and Gardner (2015)]] describe things one might do to improve their intuition.&lt;/ref&gt;

As the time since the [[w:Atomic bombings of Hiroshima and Nagasaki|atomic bombings if Hiroshima and Nagasaki]] increases, the ''intuition'' that political and military leaders have about nuclear weapons gets worse, because that history tells them that they can use more military force, even threatening to use nuclear weapons, without seriously risking a nuclear war. That intuition increasingly threatens the entirity of humanity.

=== Increasing risks with nuclear proliferation === 

Narang and Sagan, eds. (2022) ''The Fragile Balance of Terror: Deterrence in the New Nuclear Age'' includes 8 chapters by 12 authors reviewing the literature on different aspects of nuclear deterrence today. They raised many questions about the applicability of [[w:Cold War|Cold War]] analyses of deterence in an age with [[Forecasting nuclear proliferation|an increasing number of nuclear weapon states]]. They mentioned numerous concerns including the following:
* [[w:2008 Mumbai attacks|During terrorist attacks in Mumbai in 2008]], someone called called Pakistani president Zardari claiming to be Indian foreign minister Mukherjee threatening to attack Pakistan. That crises was diffused without escalation after US secretary of state Condoleezza Rice called Mukherjee, who assured her that he had not placed such a call, and India was ''not'' planning to attack Pakistan. If someone claiming to be a US official had placed a similar call to Kim Jong Un while Donald Trump was President of the US, the result may not have been as benign.&lt;ref&gt;Narang and Sagan (2022, p. 241).&lt;/ref&gt; 
* [[w:2018 Hawaii false missile alert|&quot;In January 2018, the Hawaii emergency management system issued an incoming missile warning alert]] adding, 'this is not a drill.'&quot; The US did not respond, because (a) they had redundant early warning systems that did not indicate an incoming missile, (b) professional operators in Hawaii promptly acknowledged the mistake, and (c) no one in the US seriously expected such an attack. If this had happened in North Korea, none of these three restraining conditions were present: (a) They did not have redundant warning systems. (b) Operators are killed, not just fired in North Korea for making a mistake like that. (c) US &quot;President Trump was threatening 'fire and fury' if North Korean nuclear and missile tests continued.&quot;&lt;ref&gt;Narang and Sagan (2022, p. 232).&lt;/ref&gt;
* [[w:2019 Balakot airstrike|In 2019 India bombed an alleged terrorist training camp in Balakot]], Pakistan. This was &quot;the first time a nuclear weapons state has bombed the undisputed territory of another nuclear weapons state.&quot;&lt;ref&gt;Narang and Sagan (2022, pp. 231-232).&lt;/ref&gt; 
* [[w:2020–2021 China–India skirmishes|In 2020, Chinese and Indian troops engaged in hostilities along their disputed border]] with fatalities on both sides, &quot;for the first time in almost half a century.  Intense conflict between three nuclear powers simultaneously is no longer a remote possibility.&lt;ref&gt;Narang and Sagan (2022, p. 232).&lt;/ref&gt;

Beyond this, [[w:Richard Ned Lebow|Richard Ned Lebow]] said, &quot;There’s all kinds of empirical evidence that a deterrence strategy is as likely to provoke the behavior it seeks to prevent as not.&quot;&lt;ref&gt;Lebow et al. (2023). See also Lebow (2020, ch. 4).&lt;/ref&gt;

=== System accidents ===

The concept of &quot;normal accidents&quot; or &quot;[[w:system accident|system accidents]]&quot; seems important here.  Research in that area has established that ''it is impossible to design and manage complex systems to ultra-high levels of reliability''.  Maintenance on redundant systems is often deferred, because responsible managers are often reluctant to spend money fixing something that works.&lt;ref&gt;e.g., Sagan (1993).&lt;/ref&gt; And procedures are sometimes secretly modified by people with different priorities from their management.  For example, at least between 1970 and 1974 the codes in US Air Force launch control centers for [[w:Intercontinental ballistic missile|Intercontinental ballistic missiles]] were all set continuously to 00000000.&lt;ref&gt;Ellsberg (2017, p. 61).&lt;/ref&gt; This clearly negated the claim that only the President of the US could order the use of US nuclear weapons, secured by secret codes carried in a briefcase (called the [[w:nuclear football|&quot;nuclear football&quot;]]) near the President at all times. Similarly, former US Secretary of Defense William J. Perry has said an actual nuclear attack on the US is far less likely than a report of one generated by a malfunction in the US nuclear command, control, and communications systems.&lt;ref&gt;Perry and Collina (2020).  Of course, a nuclear war could be started accidentally by any nuclear-weapons state, as suggested in the report of an Indian cruise missile that landed 2022-03-10 in Pakistan (Mashal and Masood 2022).&lt;/ref&gt;

A tragic example of a system accident is the [[w:Sinking of MV Sewol|Sinking of MV ''Sewol'']], 2014-04-16.  It sank with over twice its rated load under the command of a substitute captain. The regular captain had complained of deferred maintenance threatening the stability of the vessel;  he said the company had threatened to fire him if he continued to complain.  

As of this writing, it has been over 77 years since nuclear weapons were detonated in hostilities. As noted above, that history feeds human intuition that we can safely be more aggressive in developing, deploying and threatening the use of nuclear weapons without seriously risking [[Time to nuclear Armageddon|nuclear Armageddon]].  People who disagree like the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] with their [[w:Doomsday Clock|Doomsday Clock]] are dismissed as unrealistic, like [[w:Chicken Little|Chicken Little]].

== Human psychology and the role of the media == 

When people are attacked, it can sometimes be difficult to control their responses, which are driven by instinctive reactions often characterized as irrational.  Johnson (2004) documented how these instinctive reactions exist, because they provided survival benefits to our ancestors over hundreds of thousands and millions of years of evolutionary history.  These instincts may, however, push us into the ''worst'' possible response to a nuclear attack.   

Worse, major media everywhere have a conflict of interest in honestly reporting on anything (like these research results) that might threaten those who control the money for the media.&lt;ref name='McC+Cagé+Rolnik&quot;&gt;McChesney (2004).  Cagé (2016).  Rolnik et al. (2019). See also &quot;[[Confirmation bias and conflict]]&quot;.&lt;/ref&gt; Everyone thinks they know more than they do,&lt;ref name=Kahneman&gt;Kahneman (2011).&lt;/ref&gt; which makes them easily misled by the media they find credible.&lt;ref&gt;[[Confirmation bias and conflict]]. See also McChesney (2004), Cagé (2016), and Rolnik et al. (2019).&lt;/ref&gt;

== Probability of a nuclear war ==

The section on [[Time to nuclear Armageddon#Relevant literature|Relevant literature]] of the Wikiversity article on [[Time to nuclear Armageddon]] includes a table summarizing previous estimates of the probability of a nuclear war. Karger et al. (2023) provides a more extensive study of the probability of a nuclear war and other extistential risks. 

== Recapitulation ==

In sum, the worst possible response to a nuclear attack would seem to be a nuclear response.  

Existing nuclear weapons policies appear to be supported by propaganda that is effective, because it supports the preferences of those who control the money for the media,&lt;ref name='McC+Cagé+Rolnik&quot;/&gt; and because everyone thinks they know more than they do.&lt;ref name=Kahneman/&gt;

== Acknowledgements == 

Thanks to Owen B. Toon, Alan Robock, and presenters at their irregular webinar series on impact on climate of a nuclear war.  Of course, any errors and other deficiencies in this article are solely the responsibility of the author. 

== See also == 

* [[Expertise of military leaders and national security experts]]
* [[Time to nuclear Armageddon]] 
* [[Forecasting nuclear proliferation]] 
* [[Time to extinction of civilization]]

== References == 
* &lt;!-- Guardian (2001-10-14) Bush rejects Taliban offer to hand Bin Laden over --&gt;{{cite Q|Q111228506}}
* &lt;!-- Aleksandrov and Stenchikov (1983) &quot;On the modeling of the climatic consequences of the nuclear war&quot; --&gt;{{cite Q|Q63229964}}
* &lt;!-- Borger (2022) Five of world’s most powerful nations pledge to avoid nuclear war, Guardian --&gt;{{cite Q|Q111011203}}
* &lt;!-- Cagé (2016) Saving the media: Capitalism, crowdfunding and democracy (Harvard U. Pr.)--&gt;{{cite Q|Q54640583}} 
* &lt;!-- Chenoweth and Stephan (2011) Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict (Columbia U. Pr.) --&gt;{{cite Q|Q88725216}} For their data see, &lt;!-- Chenoweth, NAVCO data project, Harvard --&gt;{{cite Q|Q55842589}}
* &lt;!-- Coup et al. (2019) Nuclear Winter Responses to Nuclear War Between the United States and Russia in the Whole Atmosphere Community Climate Model Version 4 and the Goddard Institute for Space Studies ModelE --&gt;{{cite Q|Q111222900}}
* &lt;!-- Dikötter (2010) Mao's Great Famine (Bloomsbury) --&gt;{{cite Q|Q3209496}}
* &lt;!-- Douthat (2022) &quot;How to Stop a Nuclear War&quot;, NYT --&gt;{{cite Q|Q111145224}}
* &lt;!-- Ellsberg (2017) The Doomsday Machine: Confessions of a Nuclear War Planner (Bloomsbury) --&gt;{{cite Q|Q63862699}}
* &lt;!--Ellsberg, Goodman and González (2017) &quot;Daniel Ellsberg Reveals He was a Nuclear War Planner, Warns of Nuclear Winter &amp; Global Starvation&quot;, Democracy Now!--&gt;{{cite Q|Q64226035}}
* &lt;!-- Halimi, Serge (2018-08) &quot;The forgotten communist quarrel&quot;, Le Monde Diplomatique --&gt;{{cite Q|Q97657492}}.  
* &lt;!-- Helfand, Ira I2013) &quot;Nuclear famine: two billion people at risk?&quot;, International Physicians for the Prevention of Nuclear War --&gt;{{cite Q|Q63256454}}
* &lt;!-- Doug Hubbard (2020) The Failure of Risk Management: Why it's broken and how to fix it
Second edition (Wiley)--&gt;{{cite Q|Q123514276}}
* &lt;!-- Jägermeyr, J., et al. (2020-03-16) &quot;A regional nuclear conflict would compromise global food security&quot;, Proceedings of the National Academy of Science of the United States of America --&gt;{{cite Q|Q90371058}}
* &lt;!-- Dominic D. P. Johnson (2004). Overconfidence and War: The Havoc and Glory of Positive Illusions (Harvard U. Pr.) --&gt;{{cite Q|Q118106389}}
* &lt;!-- Johnston (2001) Chronological Listing of Above Ground Nuclear Detonations --&gt;{{cite Q|Q111222177}}
* &lt;!-- Jones, Seth, and Martin C. Libicki (2008) &quot;How Terrorist Groups End: Lessons for Countering al Qa'ida&quot;, RAND Corporation--&gt;{{cite Q|Q57515305}}
* &lt;!-- Kahneman, Daniel (2011) Thinking, Fast and Slow (FSG)--&gt;{{cite Q|Q983718}}
* &lt;!-- Kahneman and Klein (2009) Conditions for intuitive expertise: a failure to disagree--&gt;{{cite Q|Q35001791}}
* &lt;!-- Kahneman, Sibony, and Sunstein (2021) Noise:  A flaw in human judgment --&gt;{{cite Q|Q107108766}}
* &lt;!--Ezra Karger, Josh Rosenberg, Zachary G Jacobs, Molly Hickman, Rose Hadshar, Kayla Gamin, Taylor Smith, Bridget Williams, Tegan McCaslin, Stephen Thomas, and Philip Tetlock (2023) &quot;Forecasting Existential Risks: Evidence from a Long-Run Forecasting Tournament&quot;--&gt;{{cite Q|Q122208144}}
* &lt;!-- Richard Ned Lebow (2020) A Democratic foreign policy: Regaining American influence abroad (Palgrave Macmillan)--&gt;{{cite Q|Q124351867}}
* &lt;!-- Lebow, Samuelson, Graves (2023) &quot;Richard Ned Lebow on national defense including deterrence&quot;--&gt;{{cite Q|Q124351846}} 
* &lt;!-- Mujib Mashal and Salman Masood (2022-03-12) &quot;India Accidentally Fires a Missile at Pakistan. Calm Ensues.&quot;, NYT --&gt;{{cite Q|Q111223210}}
* &lt;!-- McChesney, Robert (2004) The Problem of the Media: U.S. Communication Politics in the 21st Century (Monthly Review Press) --&gt;{{cite Q|Q7758439}}
* &lt;!-- Paul E. Meehl (1954) Clinical vs. statistical prediction--&gt;{{cite Q|Q115455297}}
* &lt;!-- Narang, Vipin; Sagan, Scott D. (2022) The Fragile Balance of Terror: Deterrence in the New Nuclear Age (Cornell University Press)--&gt;{{cite Q|Q124351052|authors=Vipin Narang and Scott D. Sagan, eds.}}
* &lt;!-- Pape, Robert, and James K. Feldman (2010) Cutting the fuse : the explosion of global suicide terrorism and how to stop it (U. of Chicago Pr.)--&gt;{{cite Q|Q109249408}}
* &lt;!-- Perry, William J., and Tom Z. Collina (2020) The Button:  The new nuclear arms race and presidential power from Truman to Trump (BenBella)-&gt;&gt;{{cite Q|Q102046116}}
* &lt;!-- Robock, Alan, Luke Oman, Georgiy L. Stenchikov, Owen B. Toon, C. Bardeen, and R. P. Turco (2007) &quot;Climatic consequences of regional nuclear conflicts&quot;, Atmospheric Chemistry and Physics --&gt;{{cite Q|Q21129034}}
* &lt;!-- Robock, Alan, and Stewart C. Prager (2021-12-02) &quot;Geoscientists Can Help Reduce the Threat of Nuclear Weapons&quot;, Eos--&gt;{{cite Q|Q111146317}}
* &lt;!-- Guy Rolnik; Julia Cagé; Joshua Gans; Ellen P. Goodman; Brian G. Knight; Andrea Prat; Anya Schiffrin (1 July 2019), Protecting Journalism in the Age of Digital Platforms (PDF), Booth School of Business--&gt;{{cite Q|Q106465358}}
* &lt;!-- Sagan, Scott (1993) The Limits of Safety: Organizations, Accidents, and Nuclear Weapons (Princeton University Press)--&gt;{{cite Q|Q111146417}}
* &lt;!-- Sen, Amartya (1999) Development as Freedom (Knopf)--&gt;{{cite Q|Q5266729}}
* &lt;!--Philip E. Tetlock and Dan Gardner (2015) Superforecasting: The Art and Science of Prediction (Crown)--&gt;{{cite Q|Q21203378}}
* &lt;!-- Tegmark (2023) How would a nuclear war between Russia and the US affect you personally?--&gt;{{cite Q|Q124432900}}
* &lt;!-- Toon, Owen B., Charles G. Bardeen, Alan Robock, Hans Kristensen, Matthew McKinzie, R. J. Peterson, Cheryl S. Harrison, Nicole S. Lovenduski, and Richard P. Turco (2019) &quot;Rapidly expanding nuclear arsenals in Pakistan and India portend regional and global catastrophe&quot;, Sciences Advances--&gt;{{cite Q|Q90735736}}
* &lt;!-- Turco, R. P., Owen B. Toon, T. P. Ackerman, J. B. Pollack, and Carl Sagan (1983) &quot;Nuclear winter: Global consequences of multiple nuclear explosions&quot;, Science, 222(4630), 1283–1292, https://doi.org/10.1126/science.222.4630.1283. --&gt;{{cite Q|Q111146500}}
* &lt;!-- Turcotte (2022-03-09) Global community must step up pressure on Putin --&gt;{{cite Q|Q111235117}}
* &lt;!-- Tyler, Tom R. (2006) Why people obey the law, revised ed. (Princeton U. Pr.)--&gt;{{cite Q|Q111097755}}
* &lt;!-- Tyler, Tom R., and Yuen J. Huo (2002) Trust in the Law: Encouraging Public Cooperation with the Police and Courts (Russell Sage Foundation)--&gt;{{cite Q|Q106943244}}
* &lt;!-- Woodham-Smith, Cecil (1962) The Great Hunger: Ireland 1845-1849 (Harper)--&gt;{{cite Q|Q7737800}}
* &lt;!-- Xia et al. (2022) Global food insecurity and famine ... from a nuclear war ...--&gt;{{cite Q| Q113732668}}

== Notes ==
{{Reflist|30em}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Political science]]
[[Category:Military]]
[[Category:Military Science]]
[[Category:Freedom and abundance]]
[[Category:psychology]]
[[category:Political economy]]</text>
      <sha1>jzor62kjqjen4qbhmi164iiqdyqeror</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2810594</id>
      <parentid>2810436</parentid>
      <timestamp>2026-05-20T13:59:37Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2810594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="kiu6tapmx8pjxz4m711gp397fhxflge" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260520.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

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    <title>Social Victorians/Terminology</title>
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      <text bytes="204346" sha1="0h8tnwr48og09i3siui3rj2ijbxxw5w" xml:space="preserve">Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have.

We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the ''OED'' does.

== Articles or Parts of Clothing: Men's ==
[[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below.

=== À la Romaine ===
[[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']]
A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls.

Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments.

For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' &quot;at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey.&quot;&lt;ref&gt;Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.&lt;/ref&gt;{{rp|81, Col. 2c}}

[[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]]
[[File:Frans_Hals_-_The_Meagre_Company_(detail)_-_WGA11119.jpg|thumb|Frans Hals - The Meagre Company (detail) - WGA11119.jpg]]

=== Cavalier ===
As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier style established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.&lt;ref&gt;{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&amp;oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.&lt;/ref&gt;

Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress.

The cavalier style included gloves with large gauntlets, lace on boots, more loosely fitted breeches, coats or doublets, which were slashed so the shirt beneath was visible. Men who dressed in cavalier style also wore large and, later, powdered wigs, like those of Louis XIV, having taken the French style back to Britain.

Neck treatments in the cavalier style were falling bands, wide lace collars and jabots. These were all looser, unsupported with wires, the way the earlier ruffs were, and unstarched.

=== Coats ===

==== Doublet ====
* In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''.
* Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet.

==== Pourpoint ====
A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor.

==== Surcoat ====
Sometimes just called ''coat''.

[[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]]
=== Hose, Stockings and Tights ===
Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights.

In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses.

The ''Oxford English Dictionary'' defines tights as &quot;Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress.&quot;&lt;ref&gt;“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.&lt;/ref&gt; By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted.

''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous.

In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling.

The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait.

In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject.

=== Shoes and Boots ===

== Articles or Parts of Clothing: Women's ==

=== '''Chérusque''' ===
According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.&lt;ref&gt;{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&amp;oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.&lt;/ref&gt;

=== Corsage ===
According to the ''Oxford English Dictionary'', the corsage is the &quot;'body' of a woman's dress; a bodice.&quot;&lt;ref&gt;&quot;corsage, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.&lt;/ref&gt; This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making.

The sense of &quot;a bouquet worn on the bodice&quot; is, according to the ''OED'', American.

=== Décolletage ===

=== Girdle ===

=== Mancheron ===
According to the ''Oxford English Dictionary'', a ''mancheron'' is a &quot;historical&quot; word for &quot;A piece of trimming on the upper part of a sleeve on a woman's dress.&quot;&lt;ref&gt;&quot;mancheron, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.&lt;/ref&gt; At the present, in French, a ''mancheron'' is a cap sleeve &quot;cut directly on the bodice.&quot;&lt;ref&gt;{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&amp;oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).&lt;/ref&gt;

=== Paletot ===
A cloak or jacket worn by both women and men in different periods. In the late 19th century, we see Victoria wearing them frequently, sometimes dressed for outdoors but not always.

Paletot-redingote:&lt;blockquote&gt;United Kingdom. Introduced in 1867, ladies' fitted long coat cut without a waist seam. It had revers and buttoned down the front. They sometimes had capes.&lt;ref name=&quot;:7&quot; /&gt;{{rp|217}}&lt;/blockquote&gt;

According to the French ''Wikipédia'', a paletot is longer than hip length, has long sleeves, opens in the front.&lt;ref&gt;{{Cite journal|date=2026-02-20|title=Manteau (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manteau_(v%C3%AAtement)&amp;oldid=233467144|journal=Wikipédia|language=fr}}&lt;/ref&gt;

=== Petticoat ===

According to the ''O.E.D.'', a petticoat is a &lt;blockquote&gt;skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.&lt;ref&gt;“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, &lt;https://doi.org/10.1093/OED/1021034245&gt;&lt;/ref&gt; &lt;/blockquote&gt;This sense is, according to the ''O.E.D.'', &quot;The usual sense between the 17th and 19th centuries.&quot; However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. 

In the 19th century, women wore their chemises, bloomers and [[Social Victorians/Terminology#Hoops|hoops]] under their petticoats. 

=== Stomacher ===
According to the ''O.E.D.'', a stomacher is &quot;An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,&quot;&lt;ref&gt;“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, &lt;https://doi.org/10.1093/OED/1169498955&gt;&lt;/ref&gt; although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry.

=== Train ===
A train is

The Length of the Train

'''For the monarch [or a royal?]'''

According to Debrett's,&lt;blockquote&gt;A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer:

* Duchess: for rows of ermine; train to be six feet
* Marchioness: three and a half rows of ermine; train to be three and three-quarters feet
* Countess: three rows of ermine; train to be three and a half feet
* Viscountess: two and a half rows of ermine; train to be three and a quarter feet
* Baroness: two rows of ermine; train to be three feet&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.&lt;/ref&gt;
&lt;/blockquote&gt;The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms.

Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?''']

== Hats, Bonnets and Headwear ==

=== Women's ===
The dresses in the 1892 production of Reyer's Salammbo, based on the Flaubert novel, were influential and occasioned a lot of newspaper coverage:&lt;blockquote&gt;Among the concessions to women made recently in Paris, and over which old-fashioned folk shake their heads as being a terrible innovation, is the permission given to sit in the orchestra stalls at the theatre. Though only in the two last rows of the spectators, women of the first class had place, they are still obliged to appear in demi-toilette, which includes the wearing of a bonnet. It was on the occasion of the first performance of “Salammbo” that the change was allowed, and there are not wanting people who think that after such a departure a deluge, or some such visitation, may be looked for.&lt;ref&gt;&quot;Ladies Column.&quot; ''Kilburn Times'' 8 July 1892, Friday: 7 [of 8], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001813/18920708/175/0007. Print title: ''The Kilburn Times, Hampstead and North-Western Post'', p. 7&lt;/ref&gt;&lt;/blockquote&gt;[[Social Victorians/People/Bourke|Gwendolen Bourke]] was dressed as Salammbo at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]].

==== Fontanges ====
[[File:Recueil de modes - Tome 4 - cent-quatre-vingt-cinq planches - estampes - btv1b105296325 (083 of 195).jpg|thumb|Recueil de modes - Tome 4 - cent-quatre-vingt-cinq planches - estampes - btv1b105296325 (083 of 195).jpg]][[File:Madame de Ludre en Stenkerke et falbala - (estampe) (2e état) - N. arnoult fec - btv1b53265886c.jpg|none|thumb|Madame de Ludre en Stenkerke et falbala - (estampe) (2e état) - N. arnoult fec - btv1b53265886c.jpg]]

==== Widow's Cap ====
or mourning bonnet

According to Kate Strasdin, widow's caps were &quot;white crinkled crape [sic] objects with long streamers flowing down the back, ... customarily worn by single old women who had never remarried.&quot;&lt;ref&gt;Strasdin, Kate. ''The Dress Diary: Secrets from a Victorian Woman's Wardrobe''. Pegasus, 2023.&lt;/ref&gt;{{rp|734 of 1124}}

[[Social Victorians/People/Queen Victoria#Widow's Cap|Queen Victoria's widow's caps]] and [[Social Victorians/People/Queen Victoria#Headdresses|other headdresses]] are discussed on her page.

=== Men's ===

== Cinque Cento ==
According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.&lt;ref&gt;&quot;cinquecento, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.&lt;/ref&gt; The term, then would refer, perhaps informally, to the sixteenth century.

== Corset ==
[[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]]
The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. 

This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.)

* This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips.
* The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists.
* The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up.
* The sharp definition of the waist was caused by
** length of the corset (especially on the sides)
** the stiffness of the boning
** the layers of fabric
** the lacing (especially if the woman used tightlacing)
** the over-all shape, which was so much wider at the top and the bottom
** the contrast between the waist and the wider top and bottom
* The late-19th-century corset was long, ending below the waist even on the sides and back.
* The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage.
* The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt.
* This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels.
The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century &quot;look&quot; is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets.

==== Things To Add ====
[[File:Woman's Corset LACMA M.2007.211.353.jpg|thumb|Woman's Corset LACMA M.2007.211.353.jpg|none]]

* Corset as an outer garment, 18th century, in place of a stomacher&lt;ref name=&quot;:11&quot; /&gt;{{rp|419}}
* Corsets could be laced in front or back
* Methods for making the holes for the laces and the development of the grommet (in the 1830s)

== Court Dress ==
Also Levee and drawing-room

== Crevé ==
''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.&lt;ref&gt;&quot;creve, v.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.&lt;/ref&gt; ['''With the acute accent, it looks like a past participle in French.''']

== Elaborations ==
In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”&lt;ref name=&quot;:1&quot;&gt;Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).&lt;/ref&gt;{{rp|93}}

Looking at these elaborations as meaningful rather than dismissing them as failed attempts at &quot;historical accuracy&quot; reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,&lt;blockquote&gt;In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.&lt;ref&gt;Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&amp;A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.&lt;/ref&gt;{{rp|117}}&lt;/blockquote&gt;

[[File:Aglets from Spanish portraits - collage by shakko.jpg|thumb|alt=A collage of 12 different ornaments typically worn by elite people from Spain in the 1500s and later|Aglets — Detail from Spanish Portraits]]
=== Aglet, Aiglet ===
Historically, an aglet is a &quot;point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice.&quot;&lt;ref name=&quot;:7&quot;&gt;Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.&lt;/ref&gt;{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches).

Alonso Sánchez Coello's c. 1584&lt;ref name=&quot;:11&quot; /&gt;{{rp|316}} portrait (above right, in the [[Social Victorians/Terminology#16th Century|Hoops section]]) shows infanta Isabel Clara Eugenia wearing a vertugado, with its &quot;typically Spanish smooth cone-shaped contour,&quot; with &quot;handsome aiglets cascad[ing] down center front.&quot;&lt;ref name=&quot;:11&quot;&gt;Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper &amp; Row, 1965.&lt;/ref&gt;{{rp|315}}

=== Berthe ===
Can be spelled ''bertha''.

A wide collar made of lace and gathered at the neckline, sometimes covering the arms. Lewandowski says,&lt;blockquote&gt;Wide collar popular on women's gowns. Accented dropped shoulder line. Often made of lace.&lt;ref name=&quot;:7&quot; /&gt;{{rp|29}}&lt;/blockquote&gt;

=== Dags ===
Popular in European dress 1450–1550, dagging was a  &quot;hanging end or shred&quot; decoration on the edges of outer clothing, with a similar term used for &quot;a row of decorative strips of cloth that may ornament a tent, booth or fairground.&quot;&lt;ref&gt;{{Cite journal|date=2026-05-14|title=dag|url=https://en.wiktionary.org/w/index.php?title=dag&amp;oldid=90785397|journal=Wiktionary, the free dictionary|language=en}}&lt;/ref&gt; Often dagging would be used to hem the bottom edges of hoods, doublets, tabards and chain mail.

=== Flounce ===
A ruffle that is gathered on one edge, the bottom edge is free. Flounces are typically part of the decoration on a skirt.

=== Frou-frou ===
[[File:SarahBernhardt alsKameliendame1881.jpg|left|thumb|Bernhardt, 1881]]
In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.&lt;ref&gt;{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&amp;oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.&lt;/ref&gt; The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.&lt;ref&gt;{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&amp;oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.&lt;/ref&gt; ''Frou-frou'' is also a 1869 French drawing-room comedy by Henri Meilhac and Ludovic Halévy&lt;ref&gt;{{Cite journal|date=2025-04-19|title=Henri Meilhac|url=https://en.wikipedia.org/w/index.php?title=Henri_Meilhac&amp;oldid=1286340698|journal=Wikipedia|language=en}}&lt;/ref&gt; and performed by Sarah Bernhardt in London in 1881 (Bernhardt, left, in costume ['''conflicting info, is a photo of Bernhardt in ''La Dame aux Camélias'' instead'''?]).

''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial and, perhaps, excessive.

=== Plastics ===
Small poufs of fabric connected in a strip in the 18th century, Rococo styles.

=== Pouf, Puff, Poof ===
According to the French ''Wikipédia'', a pouf was, beginning in 1744, a &quot;kind of women's hairstyle&quot;:&lt;blockquote&gt;The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)&lt;/blockquote&gt;''Puff'' and ''poof'' are used to describe clothing.

=== Shirring ===
''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam.

=== Sequins ===
Sequins, paillettes, spangles

Sequins — or paillettes — are &quot;small, scalelike glittering disks.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|216}} The French ''Wiktionnaire'' defines ''paillette'' as &quot;Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]&quot;&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&amp;oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.&lt;/ref&gt;

According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.&lt;ref&gt;“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023,  https://doi.org/10.1093/OED/4074851670.&lt;/ref&gt; The first instance of ''spangle'' as &quot;A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts&quot; is from c. 1420.&lt;ref&gt;“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023,  https://doi.org/10.1093/OED/4727197141.&lt;/ref&gt; The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.&lt;ref name=&quot;:8&quot; /&gt;

Currently many distinguish between sequins (which are smaller) and paillettes (which are larger).

Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins.

=== Trim and Lace ===
''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:&lt;blockquote&gt;For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.&lt;ref name=&quot;:9&quot;&gt;[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]&lt;/ref&gt;{{rp|6–7}}&lt;/blockquote&gt;

==== Gold and Silver Fabric and Lace ====

The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)&lt;blockquote&gt;GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — &quot;And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work.&quot; In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel&quot; worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold &quot;tissue&quot; to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver.

Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the &quot;gold wire&quot; of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &amp;c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. &quot;Gold and silver,&quot; says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, &quot;are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly.&quot;

Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as &quot;orris,&quot; fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality.  Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies.

The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &amp;c., and the remainder is worked up in the gold lace trade.&lt;ref&gt;William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. &quot;Gold and Silver Lace.&quot; In &quot;Gold.&quot; ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.&lt;/ref&gt;&lt;/blockquote&gt;

==== Honiton Lace ====

Kate Stradsin says,&lt;blockquote&gt;Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.&lt;ref&gt;Strasdin, Kate. &quot;Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research.&quot; ''The Court Historian'' 24.2 (2019): 181-196. Rpt  http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads &quot;Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983.&quot;&lt;/ref&gt;&lt;/blockquote&gt;

[[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]]
==== Passementerie ====
''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great &quot;''exubérance''&quot; in passementerie in French design, including the development and widespread use of the Jacquard loom.&lt;ref&gt;{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&amp;oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.&lt;/ref&gt;

==== Point d'Alençon Lace ====
A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure.

Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,&lt;ref name=&quot;:7&quot; /&gt;{{rp|233}} so Alençon point is &quot;a two thread [needlepoint] lace.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|7}} Alençon lace has a &quot;floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|7}}

The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.&lt;ref&gt;{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.&lt;/ref&gt; The consistency in this sample suggests it may have been made by machine.

== Elastic ==
Elastic had been invented and was in use by the end of the 19th century. For the sense of &quot;Elastic cord or string, usually woven with india-rubber,&quot;&lt;ref name=&quot;:6&quot;&gt;“elastic, adj. &amp; n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, &lt;https://doi.org/10.1093/OED/1199670313&gt;.&lt;/ref&gt; the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: &quot;The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap.&quot;&lt;ref name=&quot;:6&quot; /&gt;

== Fabric ==

=== Brocatelle ===
Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:&lt;blockquote&gt;La '''brocatelle''' est un type de tissu datant du &lt;abbr&gt;xvi&lt;sup&gt;e&lt;/sup&gt;&lt;/abbr&gt; siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.&lt;ref&gt;{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&amp;oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.&lt;/ref&gt;&lt;/blockquote&gt;Which translates to this:&lt;blockquote&gt;Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.&lt;/blockquote&gt;The ''Oxford English Dictionary'' says, brocatelle is an &quot;imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed&quot; between the late 17th century and 1888, the last time this definition was revised.&lt;ref&gt;&quot;brocatelle, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.&lt;/ref&gt;

=== Broché ===
Lewandowski says, &quot;to be woven with a raised figure or to be embossed.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|39}} In English, the word might be spelled with or without the acute accent on the final ''e''. Generally, the term was used loosely to describe fabric with a pattern woven into it, either in the same color or a color different from that of the background. That is, the weave that produces the pattern is different from the weave that produces the background.

S. F. A. Caulfeild and B. C. Saward published this definition of ''broché'' in their 1887 ''Dictionary of Needlework'', according to the ''Oxford English Dictionary'' (the ''face'' being the side of the fabric facing the viewer):&lt;blockquote&gt;Broché. A French term denoting a velvet or silk textile, with a satin figure thrown up on the face.&lt;ref&gt;“Broché, Adj.” ''Oxford English Dictionary'', Oxford UP, December 2024, https://doi.org/10.1093/OED/1054215522.&lt;/ref&gt;&lt;/blockquote&gt;
=== Chiffon ===
A lightweight, somewhat sheer silk fabric, chiffon would have been worn only by the social elite at the end of the 19th century.&lt;ref name=&quot;:25&quot;&gt;{{Cite journal|date=2025-10-12|title=Chiffon (fabric)|url=https://en.wikipedia.org/w/index.php?title=Chiffon_(fabric)&amp;oldid=1316464288|journal=Wikipedia|language=en}}&lt;/ref&gt; Synthetic fibers were not invented until the 20th century — nylon chiffon in 1938 and polyester chiffon not until 1958.&lt;ref name=&quot;:25&quot; /&gt;

=== Ciselé ===

=== Crape ===
The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is &quot;often borrowed [from the French] as a term for all crapy fabrics other than ordinary [[Social Victorians/Mourning|black mourning crape]],&quot;&lt;ref name=&quot;:24&quot;&gt;&quot;crêpe, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.&lt;/ref&gt; with usage examples ranging from 1797 to the mid 20th century. This distinction seems more prescriptive than descriptive since texts from the 19th century to now do not make it reliably. Sometimes 19th-century newspapers put an acute accent on the ''e'' and spelled it crépe.

The fabric used for full mourning was black crape, a fabric with a dull texture, but writers continue to vary in how to spell it. Julia Baird uses ''crêpe'', defining it as &quot;a thick black rustling material made of silk, crimped to make it look dull.&quot;&lt;ref&gt;Baird, Julia. ''Victoria the Queen, an Intimate Biography of the Woman Who Ruled an Empire''. Random House, 2016. https://books.apple.com/us/book/victoria-the-queen/id953835024.&lt;/ref&gt;{{rp|584 of 1203}}

However it is spelled, crêpe is&lt;blockquote&gt;Any number of fabrics with characteristic crinkled or puckered surface.&lt;ref name=&quot;:7&quot; /&gt; (77)&lt;/blockquote&gt;

==== Crepe de Chine ====
Crêpe de chine, the ''OED'' says, is &quot;a white or other coloured crape made of raw silk.&quot;&lt;ref name=&quot;:24&quot; /&gt; Lewandowski defines it as &quot;a very lightweight, fine, plain weave silk fabric. ... Introduced in 1866, China crepe with soft, silky surface.&quot;&lt;ref name=&quot;:7&quot; /&gt; (77) 

==== Crepon de Chine ====
Crepon is a fabric heavier than the usual crape but treated like crape to be crinkly. Lewandowski says,&lt;blockquote&gt;Introduced in 1882, wool, silk, or blend fabric like very heavy crepe. ... Gay Nineties (1890–1900 C.E.). Popular in 1890s, woolen fabric creped to appear puffed between stripes [or] squares.&lt;ref name=&quot;:7&quot; /&gt; (77)&lt;/blockquote&gt;According to Lewandowski, ''crepon'' can also be another word for bustle (1865–1890 C.E. to present).&lt;ref name=&quot;:7&quot; /&gt; (77)

=== Crinoline ===
Technically, crinoline was a fabric made mostly of horsehair and sometimes linen, stiffened with starch or glue, similar to buckram today, used in men's military collars and [[Social Victorians/Terminology#Crinolines|women's foundation garments]]. Lewandowski defines crinoline as &lt;blockquote&gt;(1840–1865 C.E.). France. Originally horsehair cloth used for officers' collars. Later used for women's underskirts to support skirts. Around 1850, replaced by many petticoats, starched and boned. Around 1856, [[Social Victorians/Terminology#Crinoline Hoops|light metal cage]] was developed.&lt;ref name=&quot;:7&quot; /&gt;{{rp|78}}&lt;/blockquote&gt;The term has been used so consistently for the cage first introduced in the 1850s that held the skirt out from the body, however, that it is important to say ''crinoline cage'' or ''crinoline fabric'' or ''crinoline petticoat'' to be clear.

=== Épinglé Velvet ===
Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. 

=== Lace ===
While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]].

=== Liberty Fabrics ===

=== Lisse ===
According to the ''Oxford English Dictionary'', the term ''lisse'' as a &quot;kind of silk gauze&quot; was used in the 19th-century UK and US.&lt;ref&gt;&quot;lisse, n.1.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.&lt;/ref&gt;

=== Muslin ===
=== Satin ===
The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's &quot;insistence on new fabrics, new trimmings, new materials of every description&quot; at the beginning of his career in the mid 19th century:&lt;blockquote&gt;When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.&lt;ref name=&quot;:9&quot; /&gt;{{rp|6 in printed, 26 in digital book}}&lt;/blockquote&gt;

=== Selesia ===
According to the ''Oxford English Dictionary'', ''silesia'' is &quot;A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').&lt;ref&gt;&quot;Silesia, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.&lt;/ref&gt; It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — &quot;Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body.&quot;&lt;ref&gt;&quot;sleazy, adj.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.&lt;/ref&gt; — may be related.

=== Shot Fabric ===
According to the ''Oxford English Dictionary'', &quot;Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points.&quot;&lt;ref&gt;“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023,  https://doi.org/10.1093/OED/2977164390.&lt;/ref&gt; A shot fabric might also be made of silk and cotton fibers.

=== Tissue ===
A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft.

Historically, the term in English was used for a &quot;rich kind of cloth, often interwoven with gold or silver&quot; or &quot;various rich or fine fabrics of delicate or gauzy texture.&quot;&lt;ref&gt;“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.&lt;/ref&gt;

=== Tulle ===
In the 19th century, tulle — a very fine net — was a sheer woven tissue made of linen or silk. Tulle looms were invented in the late 18th century,&lt;ref name=&quot;:23&quot;&gt;{{Cite journal|date=2025-09-04|title=Tulle (tissu)|url=https://fr.wikipedia.org/w/index.php?title=Tulle_(tissu)&amp;oldid=228712045|journal=Wikipédia|language=fr}}&lt;/ref&gt; and the fabric &quot;first made by machine in 1768 in Nottingham.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|299}} By 1802 English tulle was recognized as higher quality than French tulle, even though the fabric is named for the French city.&lt;ref name=&quot;:23&quot; /&gt;

Tulle is still used today, but it is usually made of synthetic fabric.&lt;blockquote&gt;It is a finer textile than the textile referred to as &quot;net&quot;. ...

It can be made of various fibres, including silk, nylon, polyester and rayon. Polyester is the most common fibre used for tulle.&lt;ref&gt;{{Cite journal|date=2025-08-05|title=Tulle (netting)|url=https://en.wikipedia.org/w/index.php?title=Tulle_(netting)&amp;oldid=1304416320|journal=Wikipedia|language=en}}&lt;/ref&gt;&lt;/blockquote&gt;Victorian silk tulle would not have been stiff unless it was treated with sizing.

== Fan ==

The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)&lt;blockquote&gt;FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century &lt;small&gt;B&lt;/small&gt;.&lt;small&gt;C&lt;/small&gt;. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day.

Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies.

Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals.

Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously  proposes to establish a regular drill for these purposes.

The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century.

The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes.

During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other &quot;genre&quot; painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &amp;c., have been lately bequeathed to the South Kensington Museum.

Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.&lt;ref&gt;J. H. Pollen [J.H.P.]. &quot;Fan.&quot; ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.&lt;/ref&gt;&lt;/blockquote&gt;Folding fans were available and popular early and are common accessories in portraits of fashionable women through the centuries.

== Costumes for Theatre and Fancy Dress ==
Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe and North America during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. 

At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. 

=== Fancy Dress ===
In her ''Magnificent Entertainments: Fancy Dress Balls of Canada's Governors General, 1876-1898'', Cynthia Cooper describes the resources available to those needing help making a costume for a fancy-dress ball:&lt;blockquote&gt;There were a number of places eager ballgoers could turn for assistance and inspiration. Those with a scholarly bent might pore over history books or study pictures of paintings or other works of art. For more direct advice, one could turn to the barrage of published information specifically on fancy dress. Women’s magazines such as ''Godey’s Lady’s Book'' and ''The Englishwoman’s Domestic Magazine'' sometimes featured fancy dress designs and articles, and enticing specialized books were available with extensive recommendations for choosing fancy dress. By far the most complete sources were the books by [[Social Victorians/People/Ardern Holt|Ardern Holt]], a prolific British authority on the subject. Holt’s book for women, ''Fancy Dresses Described, or What to Wear at Fancy Balls'' (published in six editions between 1879 and 1896), began with the query, ‘‘But what are we to wear?” Holt’s companion book, ''Gentlemen’s Fancy Dress:'' ''How to Choose It'', was also published in six editions from 1882 to 1905. Other prominent authorities included Mrs. Aria’s ''Costume: Fanciful, Historical, and Theatrical'' and, in the US, the Butterick Company’s ''Masquerade and Carnival: Their Customs and Costumes''. The Butterick publication relied heavily on Holt, copying large sections of the introduction outright and paraphrasing other sections.&lt;ref name=&quot;:16&quot;&gt;Cooper, Cynthia. ''Magnificent entertainments: fancy dress balls of Canada's Governors General, 1876-1898''.Fredericton, N.B.; Hull, Quebec: Goose Lane Editions and Canadian Museum of Civilization, 1997. Internet Archive https://archive.org/details/magnificententer0000coop/.&lt;/ref&gt;{{rp|28–29}}&lt;/blockquote&gt;

Cynthia Cooper discusses how &quot;historical accuracy&quot; works in historical fiction and historical dress: &lt;blockquote&gt;A seemingly accurate costume and coiffure bespoke a cultured individual whose most gratifying compliment would be “historically correct.” Those who were fortunate enough to own actual clothing from an earlier period might wear it with pride as a historical relic, though they would generally adapt or remake it in keeping with the aesthetics of their own period. Historical accuracy was always in the eye of beholders inclined to overlook elements of current fashion in a historical costume. Theatre had long taught the public that if a costume appeared tasteful and attractive, it could be assumed to be accurate. Even at Queen Victoria’s fancy dress balls, costume silhouette was always far more like the fashionable dress of the period than of the time portrayed. For this reason, many extant eighteenth-century dresses show evidence of extensive alterations done in the nineteenth century, no doubt for fancy dress purposes.&lt;ref name=&quot;:16&quot; /&gt;{{rp|25}}&lt;/blockquote&gt;

The newspaper ''The Queen'' published dress and fashion information and advice under the byline of [[Social Victorians/People/Ardern Holt|Ardern Holt]], who regularly answered questions from readers about fashion as well as about fancy dress. Holt also wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. 

In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: &quot;Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known.&quot;&lt;ref&gt;Holt, Ardern. &quot;Fancy Dress a la Mode.&quot; The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.&lt;/ref&gt; Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy.

As Leonore Davidoff says, &quot;Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|93}} [handled under [[Social Victorians/Terminology#Elaborations|Elaborations]]]

=== Historical Accuracy ===
Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages.

The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor.

By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric).

* clothing construction and flat-pattern techniques
* Generally, the further back in time we go, the more 2-dimensional the clothing itself was.

==== Women's Versions of Historical Accuracy at the Ball ====
As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look &quot;Victorian&quot; to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many &quot;elaborations&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|93}}, mostly in the decorations, trim and accessories. 

Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. 

==== Men's Versions of Historical Accuracy at the Ball ====
Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways.

* Trim
* Mixing neck treatments
* Hair
* Breeches
* Shoes and boots
* Military uniforms, arms, gloves, boots

== Feathers and Plumes ==

=== Aigrette ===
Elizabeth Lewandowski defines ''aigrette'' as &quot;France. Feather or plume from an egret or heron.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|5}} Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle.

=== Prince of Wales's Feathers or White Plumes ===
The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]].

For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes.

First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that&lt;blockquote&gt;It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.

White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.&lt;ref&gt;{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.&lt;/ref&gt;&lt;/blockquote&gt;[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.&lt;ref&gt;&quot;Abstract&quot; for Blackwell, Caitlin. &quot;'&lt;nowiki/&gt;''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776.&quot; ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).&lt;/ref&gt;

Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet&lt;ref&gt;{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&amp;oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.&lt;/ref&gt; since King Edward III (1312–1377&lt;ref&gt;{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&amp;oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.&lt;/ref&gt;).

Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?'''

Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]:

# Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]]
# [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire
# Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]]
# Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]]
# Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]]
# Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]]
# Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]]
# Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]]
#[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph)
#Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay
#Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'')
#[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia
#Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise
#Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson
#[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney
#[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV
#Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761
#Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers
#Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh
#[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775
#Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes
#[[Social Victorians/People/Suffolk|Marguerite Hyde &quot;Daisy&quot; Leiter]] (at 684), wearing what looks like Prince of Wales's plumes
#Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia
#Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick
#Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri
#Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette

More men than women were wearing plumes reminiscent of the Prince of Wales's feathers:

*

==== Bibliography for Plumes and Prince of Wales's Feathers ====
* Blackwell, Caitlin. &quot;'''The Feather'd Fair in a Fright'&lt;nowiki/&gt;'': The Emblem of the Feather in Graphic Satire of 1776.&quot; Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x.
* &quot;Prince of Wales's feathers.&quot; ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page''']
* Simpson, William. &quot;On the Origin of the Prince of Wales' Feathers.&quot; ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&amp;view=1up&amp;seq=643&amp;q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress.

== Honors ==

=== The Bath ===
The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion)

=== The Garter ===
The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion)

[[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 &quot;Last Knight&quot; exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]]
=== The Golden Fleece ===
To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be &quot;the most prestigious and historic order of chivalry in the world&quot; because of its long history and strict limitations on membership.&lt;ref name=&quot;:10&quot;&gt;{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&amp;oldid=980340875|journal=Wikipedia|language=en}}&lt;/ref&gt; The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,&lt;ref name=&quot;:12&quot;&gt;Thompson, R[obert]. H[ugh]. &quot;The Golden Fleece in Britain.&quot; Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).&lt;/ref&gt; the first Protestant to be admitted to the order.&lt;ref name=&quot;:10&quot; /&gt; Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.&lt;ref name=&quot;:10&quot; /&gt;

[[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]]
The photograph (upper right) is of a Polish badge dating from the &quot;turn of the XV and XVI centuries.&quot;&lt;ref&gt;{{Citation|title=Polski:  Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.&lt;/ref&gt; The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]].

The badges and collars that Knights of the Order actually wore vary quite a bit.

The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.&lt;ref&gt;Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.&lt;/ref&gt;

=== Royal Victorian Order ===
(GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member)

=== St. John ===
The Order of the Knights of St. John

=== Star of India ===
Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion)

=== Thistle ===
The Most Ancient and Most Noble Order of the Thistle

== Hoops ==
'''This section is under construction right now'''.

Terms: farthingale, panniers, hoops, crinoline, cage, bustle

Between 1450 and 1550 a loosely woven, very stiff fabric made from linen and horsehair was used in &quot;horsehair petticoats.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|137}} Heavy and scratchy, these petticoats made the fabric of the skirt lie smooth, without wrinkles or folds. Over time, this horsehair fabric was used in several kinds of objects made from fabric, like hats and padding for poufs, but it is best known for its use in the structure of hoops, or cages. Horsehair fabric was used until the mid-19th century, when it  was called ''crinoline'' and used for petticoats again (1840–1865).&lt;ref name=&quot;:7&quot; /&gt;{{rp|78}} We still call this fabric ''crinoline''.

''Hoops'' is a mid-19th-century term for a cage-like structure worn by a woman to hold her skirts away from her body. The term ''cage'' is also 19th century, and ''crinoline'' is sometimes used in a non-technical way for 19th-century cages as well. Both these terms are commonly used now for the general understructure of a woman's skirts, but they are not technically accurate for time periods before the 19th century.

As fashion, that cage-like structure was the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat, and created the fashionable silhouette from the 15th through the late 19th century. The 16th-century Katherine of Aragon is credited with making hoops popular outside Spain for women of the elite classes. By the end of the 16th century France had become the arbiter of fashion for the western world, and it still is. The cage is notable for how long it lasted in fashion and for its complex evolution.

Together with the [[Social Victorians/Terminology#Corsets|corset]], the cage enabled all the changes in fashionable shapes, from the extreme distortions of 17th-and-18th-century panniers to the late 19th-century bustle. Early hoops circled the body in a bell, cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear as a bustle. That is, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, especially below the waist, while corsets did their work above it.

When hoops were first introduced in the 15th century, women's shoes for the first time differentiated from men's and became part of the fashionable look. In the periods when the skirts were flat in front (with the farthingale and in the transitional 17th century), they did not touch the floor, making shoes visible — and important fashion accessories. Portraits of high-status, high-fashion women consistently show their pointy-toed shoes, which would have been more likely to show when they were moving than when they were standing still. The shoes seem to draw attention to themselves in these portraits, suggesting that they were important to the painters and, perhaps, the women as well.

In addition to the shape, the materials used to make hoops evolved — from cane and wood to whalebone, then steel bands and wire. Initially fabric strips, tabs or ribbons were the vertical elements in the cages and evolved into channels in a linen, muslin or, later, crinoline underskirt encasing wires or bands. Fabrics besides crinoline —  like cotton, silk and linen — were used to connect the hoops and bands in cages. All of these materials used in cages had disadvantages and advantages. 

=== Disadvantages and Advantages ===
Hoops affected the way women were able to move. ['''something about riding'''?]

==== Disadvantages ====
the weight, getting through doorways, sitting, the wind, getting into carriages, what the dances involved. Raising '''one's''' skirts to climb stairs or walk was more difficult with hoop. 

['''Contextualize with dates?'''] &quot;The combination of corset, bustle, and crinolette limited a woman's ability to bend except at the hip joint, resulting in a decorous, if rigid, sense of bearing.&quot;&lt;ref&gt;Koda, Harold. ''Extreme Beauty: The Body Transformed.'' The Metropolitan Museum of Art, 2001.&lt;/ref&gt; (130) 

As caricatures through the centuries makes clear, one disadvantage hoops had is that they could be caught by the wind, no matter what the structure was made of or how heavy it was. 

In her 1941 ''Little Town on the Prairie'', Laura Ingalls Wilder writes a scene in which Laura's hoops have crept up under skirts because of the wind. Set in 1883,&lt;ref&gt;Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.&lt;/ref&gt; this very unusual scene shows a young woman highly skilled at getting her hoops back down without letting her undergarments show. The majority of European and North American women wore hoops in 1883, but to our knowledge no other writer from this time describes any solution to the problem of the wind under hoops or, indeed, a skill like Laura's. &lt;blockquote&gt;“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be.

“As she and Carrie hurried on she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again.

“Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.”

“They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”&lt;ref&gt;Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.&lt;/ref&gt;&lt;/blockquote&gt;The 16-year-old Laura makes the comment that she wants to be in style, but she lives on the prairie in the U.S., far from a large city, and would not necessarily wear the latest Parisian style, although she reads the American women's domestic and fashion monthly ''[[Social Victorians/Newspapers#Godey's Lady's Book|Godey's Lady's Book]]'' and would know what was stylish.

==== '''Advantages''' ====
The '''weight''' of hoops was somewhat corrected over time with the use of steel bands and wires, as they were lighter than the wood, cane or whalebone hoops, which had to be thick enough to keep their shape and to keep from breaking or folding under the weight of the petticoats and skirts. Full skirts made women's waists look smaller, whether by petticoats or hoops. Being fashionable, being included among the smart set.

The hoops moved the skirts away from the legs and feet, making moving easier. 

By moving the heavy petticoats and skirts away from their legs, hoops could actually give women's legs and feet more freedom to move.

Because so few fully constructed hoop foundation garments still exist, we cannot be certain of a number of details about how exactly they were worn. For example, the few contemporary drawings of 19th-century hoops show bloomers beneath them but no petticoats. However, in the cold and wind (and we know from Laura Ingalls Wilder how the wind could get under hoops), women could have added layers of petticoats beneath their hoops for warmth.[[File:Chaise à crinolines.jpg|thumb|Chaise à Crinolines, 19th century]]
=== Accommodation ===
Hoops affected how women sat, and furniture was developed specifically to accommodate these foundation structures. The ''chaise à crinolines'' or chair for hoop skirts (right), dating from the 2nd half of the 19th century, has a gap between the seat and the back of the chair to keep a woman's undergarments from showing as she sat, or even seated herself, and to reduce wrinkling of the fabric by accommodating her hoops, petticoats and skirts.[[File:Vermeer Lady Seated at a Virginal.jpg|thumb|Vermeer, Lady Seated at a Virginal|left]]Vermeer's c. 1673 ''Lady Seated at a Virginal'' (left) looks like she is sitting on this same kind of chair, suggesting that furniture like this had existed long before the 19th century. Vermeer's painting shows how the chair could accommodate her hoops and the voluminous fabric of her skirts.

The wide doorways between the large public rooms in the Palace of Versailles could accommodate wide panniers. '''Louis XV and XVI of France occupied an already-built Versailles, but they both renovated the inside over time'''.

Some configurations of hoops permitted folding, and of course the width of the hoops themselves varied over time and with the evolving styles and materials.

With hoops, skirts were lifted away from the legs and feet, and when skirts got shorter, to above the floor, women's feet had nearly unrestricted freedom to move. Evening gowns, with trains, were still restrictive.

A modern accommodation are the leaning boards developed in Hollywood for women wearing period garments like corsets and long, full skirts. The leaning boards allow the actors to rest without sitting and wrinkling their clothes.[[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|alt=Old oil painting of a woman wearing a dress from the 1400s holding the decapitated head of a man with a halo before a table of people at a dinner party|Pedro García de Benabarre, Detail from St. John Altarpiece, Showing Visible Hoops]]
=== Early Hoops ===
Hoops first appeared in Spain in the 15th century and influenced European fashion for at least 3 centuries.

A detail (right) from Pedro García de Benabarre's c. 1470 larger altarpiece painting shows women wearing a style of hoops that predates the farthingale but marks the beginning point of the development of that fashion. Salomé (holding John the Baptist's head) is wearing a dress with what looks like visible wooden hoops attached to the outside of the skirt, which also appears to have padding at the hips underneath it.

The clothing and hairstyles of the people in this painting are sufficiently realistic to offer details for analysis. The foundation garments the women are wearing are corsets and bum rolls. Because none still exist, we do not know how these hoops attached to the skirts or how they related structurally to the corset. The bottom hoop on Salomé's skirt rests on the ground, and her feet are covered. The women near her are kneeling, so not all their hoops show. 

The painter De Benabarre was &quot;active in Aragon and in Catalonia, between 1445–1496,&quot;&lt;ref&gt;{{Cite web|url=https://www.mfab.hu/artworks/10528/|title=Saint Peter|website=Museum of Fine Arts, Budapest|language=en-US|access-date=2024-12-11}} https://www.mfab.hu/artworks/10528/.&lt;/ref&gt; so perhaps he saw the styles worn by people like Katharine of Aragon, whose hoops are now called a farthingale.

=== Early Farthingale ===
In the 16th century, the foundation garment we call ''hoops'' was called a ''farthingale''. Elizabeth Lewandowski says that the metal supports (or structure) in the hoops were made of wire:&lt;blockquote&gt;''&quot;FARTHINGALE: Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes.&quot;&lt;ref name=&quot;:7&quot; /&gt;''{{rp|105}}&lt;/blockquote&gt;The French term for ''farthingale'' is ''vertugadin'' — &quot;un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England].&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&amp;oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.&lt;/ref&gt; The French also called the farthingale a &quot;''cachenfant'' for its perceived ability to hide pregnancy,&quot;&lt;ref&gt;&quot;Clothes on the Shakespearean Stage.&quot; Carleton Production. Amazon Web Services. https://carleton-wp-production.s3.amazonaws.com/uploads/sites/84/2023/05/Clothes-on-the-Shakespearean-Stage_-1.pdf (retrieved April 2025).&lt;/ref&gt; not unreasonable given the number of portraits where the subject wearing a farthingale looks as if she might be pregnant. The term in Spanish is ''vertugado''. Nowadays clothing historians make clear distinctions among these terms, especially farthingale, bustle and hip roll, but the terminology then did not need to distinguish these garments from later ones.&lt;p&gt;&lt;/p&gt;

The hoops on the outsides of the skirts in the Pedro García de Benabarre painting (above right) predate what would technically be considered a vertugado.[[File:Alonso Sánchez Coello 011.jpg|thumb|alt=Old painting of a princess wearing a richly jeweled outfit|Alonso Sánchez Coello, Infanta Isabel Clara Eugenia Wearing a Vertugado, c. 1584]]

Blanche Payne says,&lt;blockquote&gt;Katherine of Aragon is reputed to have introduced the Spanish farthingale ... into England early in the [16th] century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth.&lt;ref name=&quot;:11&quot; /&gt;{{rp|291}}&lt;/blockquote&gt;

In fact, &quot;The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501].&quot;&lt;ref name=&quot;:0&quot; /&gt; Catherine of Aragon, of course, married Henry VIII after Arthur's death, then was divorced and replaced by Anne Boleyn.

Of England, Lewandowski says that &quot;Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|291}} That is, hoops were &quot;quite generally worn&quot; among the ruling and aristocratic classes in England, and may have been worn by some women among the wealthy bourgeoisie. Sumptuary laws addressed &quot;certain features of garments that are decorative in function, intended to enhance the silhouette&quot;&lt;ref&gt;{{Cite journal|date=2025-02-22|title=Sumptuary law|url=https://en.wikipedia.org/wiki/Sumptuary_law|journal=Wikipedia|language=en}}&lt;/ref&gt; and signified wealth and status, but they were generally not very successful and not enforced well or consistently. (Sumptuary laws &quot;attempted to regulate permitted consumption, especially of clothing, food and luxury expenditures&quot;&lt;ref&gt;{{Cite journal|date=2024-09-27|title=sumptuary law|url=https://en.wiktionary.org/wiki/sumptuary_law|journal=Wiktionary, the free dictionary|language=en}}&lt;/ref&gt; in order to mark class differences and, for our purposes, to use fashion to control women and the burgeoning middle class.)

The Spanish vertugado shaped the skirt into an symmetrical A-line with a graduated series of hoops sewn to an undergarment. Alonso Sánchez Coello's c. 1584&lt;ref name=&quot;:11&quot; /&gt;{{rp|316}} portrait (right) shows infanta Isabel Clara Eugenia wearing a vertugado, with its &quot;typically Spanish smooth cone-shaped contour.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|315–316}}

The shoes do not show in the portraits of women wearing the Spanish cone-shaped vertugado. The round hoops stayed in place in front, even though the skirts might touch the floor, giving the women's feet enough room to take steps. 

By the end of the 16th century the French and Spanish farthingales had evolved separately and were no longer the same garment.[[File:Queen Elizabeth I ('The Ditchley portrait') by Marcus Gheeraerts the YoungerFXD.jpg|thumb|alt=Old oil painting of a queen in a white dress with shoulders and hips exaggerated by her dress|Marcus Gheeraerts the Younger, Queen Elizabeth I in a French Cartwheel Farthingale, 1592|left]]

The French vertugadin — a cartwheel farthingale — was a flat &quot;platter&quot; of hoops worn below the waist and above the hips. Once past the vertugadin, the skirt fell straight to the floor, into a kind of asymmetrical drum shape that was balanced by strict symmetry in the rest of the garment. The English Queen Elizabeth I is wearing a French drum-shaped farthingale in Marcus Gheeraerts the Younger's c. 1592 portrait (left).[[File:Hardwick Hall Portrait of Elizabeth I of England.jpg|thumb|Hilliard, Hardwick Hall Portrait of Elizabeth I of England, c. 1598–1599]]In Nicholas Hilliard's c. 1598–1599 portrait of Queen Elizabeth I (right), an extraordinary showing of jewels, pearls and embroidery from the top of her head to the tips of her toes make for a spectacular outfit. The drum of the cartwheel farthingale is closer to the body beneath the point of the bodice, and the underskirt is gathered up the sides of the foundation corset to where her natural waistline would be. The gathers flatten the petticoat from the point to the hem, and the fabric collected at the sides falls from the edge of the drum down to her ankles.

Associated with the cartwheel farthingale was a very long waist and a skirt slightly shorter in the front. A rigid corset with a point far below the waist and the downward-angled farthingale flattened the front of the skirt. Because the skirt in front over a cartwheel farthingale was closer to the woman's body and did not touch the floor, the dress flowed and the women's shoes showed as they moved. Almost all portraits of women wearing cartwheel farthingales show the little pointy toes of their shoes. In Gheeraerts' painting, Queen Elizabeth's feet draw attention to themselves, suggesting that showing the shoes was important.

Farthingales were heavy, and together with the rigid corsets and the construction of the dress (neckline, bodice, sleeves, mantle), women's movement was quite restricted. Although their feet and legs had the freedom to move under the hoops, their upper bodies were held in place by their foundation garments and their clothing, the sleeves preventing them from raising their arms higher than their shoulders. This restriction of the movement of their arms can be seen in Elizabethan court dances that included clapping. They clapped their hands beside their heads rather than over their heads.

The steady attempts in the sumptuary laws to control fine materials for clothing reveals the interest middle-class women had in wearing what the cultural elite were wearing at court.

=== The Transitional 17th Century ===
What had been starched and stiff in women's dress in the 16th century — like ruffs and collars — became looser and flatter in the 17th. This transitional period in women's clothing also introduced the [[Social Victorians/Terminology#Cavalier|Cavalier style of men's dress]], which began with the political movement in support of England's King Charles II while he was still living in France. Like the ones women wore, men's ruffs and collars were also no longer starched or wired, making them looser and flatter as well.

For much of the 17th century — beginning about 1620, according to Payne — skirts were not supported by the cage-like hoops that had been so popular.&lt;ref name=&quot;:11&quot; /&gt;{{rp|355}} Without structures like hoops, skirts draped loosely to the floor, but they did not fall straight from the waist. Except for dressing gowns (which sometimes appear in portraiture in spite of their informality), the skirts women wore were held away from the body by some kind of padding or stiffened roll around the waist and at the hips, sometimes flat in front, sometimes not. The skirts flowed from the hips, either straight down or in an A-line depending on the cut of the skirt.
[[File:The Vanity of Women Masks and Bustles MET DT4982.jpg|thumb|Maerten de Vos, ''The Vanity of Women: Masks and Bustles'', c. 1600]]
==== Hip Rolls ====
This c. 1600 Dutch engraving attributed to Maerten de Vos (right) shows two servants dressing two wealthy women in masks and hip rolls. In its title of this engraving the Metropolitan Museum of Art calls a hip roll a ''bustle'' (which it defines as a padded roll or a French farthingale),&lt;ref&gt;De Vos, Maerten. &quot;The Vanity of Women: Masks and Bustles.&quot; Metropolitan Museum of Art. Wikimedia Commons 

https://commons.wikimedia.org/wiki/File:The_Vanity_of_Women_Masks_and_Bustles_MET_DT4982.jpg.&lt;/ref&gt; but the engraving itself calls it a ''cachenfant''.&lt;ref name=&quot;:20&quot;&gt;De Vos, Maerten (attrib. to). &quot;The Vanity of Women: Masks and Bustles.&quot; Circa 1600. ''The Costume Institute: The Metropolitan Museum of Art''. Object Number: 2001.341.1. https://www.metmuseum.org/art/collection/search/82615&lt;/ref&gt; The craftsmen in the back are wearing masks. The one on the left is making the masks that the shop sells, and the one on the right is making the hip rolls.

The serving woman on the left is fitting a mask on what is probably her mistress. The kneeling woman on the right is tying a hip roll on what is probably hers.

The text around the engraving is in French and Dutch. The French passages read as follows (clockwise from top left), with the word ''cachenfant'' (farthingale) bolded:&lt;blockquote&gt;
Orne moy auecq la masque laide orde et sale:
&lt;br&gt;Car laideur est en moy la beaute principale.

Achepte dame masques &amp; passement:
&lt;br&gt;Monstre vostre pauvre [?] orgueil hardiment.

Venez belles filles auecq fesses maigres:
&lt;br&gt;Bien tost les ferayie rondes &amp; alaigres.

Vn '''cachenfant''' come les autres me fault porter:
&lt;br&gt;Couste qu'il couste; le fol la folle veult aymer.

Voy cy la boutiquel des enragez amours,
&lt;br&gt;De vanite, &amp; d'orgueil &amp; d'autres tels tours:

D'ont plusieurs qui parent la chair puante,
&lt;br&gt;S'en vont auecq les diables en la gehenne ardante.

&lt;ref name=&quot;:20&quot; /&gt;&lt;/blockquote&gt;

Which translates, roughly, into
&lt;blockquote&gt;
Adorn me with the ugly, dirty, and orderly mask:
&lt;br&gt;For ugliness is the principal beauty in me.

Buy, lady, masks and trimmings:
&lt;br&gt;Boldly show your poor [?] pride.

Come, beautiful girls with thin buttocks:
&lt;br&gt;Soon, make them round and cheerful.

I must wear a [farthingale, lit. &quot;hide child&quot;] like the others:
&lt;br&gt;No matter how much it costs; the madman wants to love.

See here the store of rabid loves,
&lt;br&gt;Of vanity, and pride, and other such tricks:

Many of whom adorn the stinking flesh,
&lt;br&gt;Go with the devils to the burning hell.
&lt;/blockquote&gt;Later versions of hoops were also used to hide or at least de-emphasize pregnancy (see [[Social Victorians/Terminology#Crinoline Hoops|Crinoline Hoops]], below).[[File:The Vanity of Women Masks and Bustles MET DT4982 (detail of padded rolls or French farthingales).jpg|thumb|Detail of Maerten de Vos, ''The Vanity of Women: Masks and Bustles'', c. 1600]]
Traditionally thought of as padding, the hip rolls, at least in this detail of the c. 1600 engraving (right), are hollow and seem to be made cylindrical by what looks like rings of cane or wire sewn into channels. The kneeling woman is tying the strings that attach the hip roll, which is being worn above the petticoat and below the overskirt that the mistress is holding up and back. The hip roll under construction on the table looks hollow, but when they are finished the rolls look padded and their ends sewn closed.

Farthingales were more complex than is usually assumed. Currently, ''farthingale'' usually refers to the cane or wire foundation that shaped the skirt from about 1450 to 1625, although the term was not always used so precisely. Padding was sometimes used to shape the skirt, either by itself or in addition to the cartwheel and cone-shaped foundational structures. The padding itself was in fact another version of hoops that were structured both by rings as well as padding. Called a bustle, French farthingale, cachenfant, bum barrel&lt;ref name=&quot;:7&quot; /&gt;{{rp|42}} or even (quoting Ben Jonson, 1601) bum roll&lt;ref&gt;Cunnington, C. Willett (Cecil Willett), and Phillis Cunnington. ''Handbook of English Costume in the Sixteenth Century''. Faber and Faber, 1954. Internet Archive https://archive.org/details/handbookofenglis0000unse_e2n2/.&lt;/ref&gt;{{rp|161}} in its day, the hip roll still does not have a stable name. The common terms for what we call the hip roll now include ''bum roll'' and ''French farthingale''. The term ''bustle'' is no longer associated with the farthingale.

==== Bunched Skirts or Padding ====
The speed with which trends in clothing changed began to accelerate in the 17th century, making fashion more expensive and making keeping up with the latest styles more difficult. Part of the transition in this century, then, is the number of silhouettes possible for women, including early forms of what became the pannier in the 18th century and what became the bustle in the late 19th. In the later periods, these forms of hoops involved &quot;baskets&quot; or cages (or crinolines), but during this transitional period, these shapes were made from &quot;stiffened rolls [&lt;nowiki/&gt;[[Social Victorians/Terminology#Hip Rolls|hip rolls]]] that were tied around the waist&quot;&lt;ref&gt;Bendall, Sarah A. () The Case of the “French Vardinggale”: A Methodological Approach to Reconstructing and Understanding Ephemeral Garments, ''Fashion Theory'' 2019 (23:3), pp. 363-399, DOI: [[doi:10.1080/1362704X.2019.1603862|10.1080/1362704X.2019.1603862]].&lt;/ref&gt;{{rp|369}} at the hips under the skirts or from bunched fabric, or both. The fabric-based volume in the back involved the evolution of an overskirt, showing more and more of the underskirt, or [[Social Victorians/Terminology#Petticoat|petticoat]], beneath it. This development transformed the petticoat into an outer garment.[[File:Princess Teresa Pamphilj Cybo, by Jacob Ferdinand Voet.jpg|thumb|Attr. to Voet, Anna Pamphili, c. 1670]]
[[File:Caspar Netscher - Girl Standing before a Mirror - 1925.718 - Art Institute of Chicago.jpg|thumb|Netscher, Girl Standing before a Mirror|left]]

Two examples of the bunched overskirt can be seen in Caspar Netscher's ''Girl Standing before a Mirror'' (left) and Voet's ''Portrait of Anna Pamphili'' (right), both painted about 1670. (This portrait of Anna Pamphili and the one below right were both misidentified with her mother Olimpia Aldobrandini.) In both these portraits, the overskirt is split down the center front, pulled to the sides and toward the back and stitched (probably) to keep the fabric from falling flat. The petticoat, which is now an outer garment, hangs straight to the floor. In Netscher's portrait, the girl's shoe shows, but the skirt rests on the ground, requiring her to lift her skirts to be able to walk, not to mention dancing. The dress in Anna Pamphili's portrait is an interesting contrast of soft and hard. The embroidery stiffens the narrow petticoat, suggesting it might have been a good choice for a static portrait but not for moving or dancing.

Besides bunched fabric, the other way to make the skirts full at the hips was with hip rolls. Mierevelt's 1629 Portrait of Elizabeth Stuart (below, left) shows a split overskirt, although the fabric is not bunched or draped toward the back. The fullness here is caused by a hip roll, which adds fullness to the hips and back, leaving the skirts flat in front. In this case the flatness of the roll in front pulls the overskirt slightly apart and reveals the petticoat, even this early in the century. One reason this portrait is striking because Elizabeth Stuart appears to be wearing a mourning band on her left arm. Also striking are the very elaborate trim and decorations, displaying Stuart's wealth and status, including the large ornament on the mourning band. [[File:Michiel van Mierevelt - Portrait of Elizabeth Stuart (1596-1662), circa 1629.jpg|thumb|Michiel van Mierevelt, Elizabeth Stuart, c. 1629|left]][[File:Attributed to Voet - Portrait of Anna Pamphili, misidentified with her mother Olimpia Aldobrandini.jpg|thumb|Attr. to Voet, Anna Pamphili, c. 1671]]

The c. 1671 portrait of Anna Pamphili (below, right) shows an example of the petticoat's development as an outer garment. In the Mierevelt portrait (left), the petticoat barely shows. A half century later, in the portrait of Anna Pamphili, the overskirt is not split but so short that the petticoat is almost completely revealed. A hip roll worn under both the petticoat and the overskirt gives her hips breadth. The petticoat is gathered at the sides and smooth in the front, falling close to her body. The fullness of the petticoat and the overskirt is on the sides — and possibly the back. The heavily trimmed overskirt is stiff but not rigid. Anna Pamphili's shoe peeps out from under the flattened front of the petticoat.

The neckline, the hipline, the bottom of the overskirt, the trim at the hem of the petticoat and overskirt and the ribbons on the sleeves — as well as even the hair style — all give Pamphili's outfit a sophisticated horizontal design, a look that soon would become very important and influential as panniers gained popularity.

=== Panniers ===
The formal, high-status dress we most associate with the 18th century is the horizontal style of panniers, the hoops at the sides of the skirt, which is closer to the body in front and back. Popular in the mid century in France, panniers continued to dominate design in court dress in the U.K. &quot;well into the 19th century.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}} ''Paniers anglais'' were 8-hoop panniers.&lt;ref name=&quot;:7&quot; /&gt;{{rp|219}}

Panniers were made from a variety of materials, most of which have not survived into the 21st century, and the most common materials used panniers has not been established. Lewandowski says that skirts were &quot;stretched over metal hoops&quot; that &quot;First appear[ed] around 1718 and [were] in fashion [for much of Europe] until 1800. ... By 1750 the one-piece pannier was replaced by [two pieces], with one section over each hip.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|219}} According to Payne, another kind of pannier &quot;consisted of a pair of caned or boned [instead of metal] pouches, their inner surfaces curved to the ... contour of the hips, the outside extending well beyond them.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|428}} Given that it is a natural material, surviving examples of cane for the structure of panniers are an unexpected gift, although silk, linen and wool also occasionally exists in museum collections. No examples of bone structures for panniers exist, suggesting that bone is less hardy than cane. Waugh says that whalebone was the only kind of &quot;bone&quot; (it was actually cartilage, of course) used;&lt;ref name=&quot;:19&quot;&gt;Waugh, Norah. ''Corsets and Crinolines''. New York, NY: Theatre Arts Books, 1954. Rpt. Routledge/Theatre Arts Books, 2000.&lt;/ref&gt;{{rp|167}} Payne says cane and whalebone were used for panniers.&lt;ref name=&quot;:11&quot; /&gt;{{rp|426}} Neither Payne nor Waugh mention metal. Examples of metal structures for panniers have also not survived, perhaps because they were rare or occurred later, during revolutionary times, when a lot of things got destroyed.

The pannier was not the only silhouette in the 18th century. In fact, the speed with which fashion changed continued to accelerate in this century. Payne describes &quot;Six basic forms,&quot; which though evolutionary were also quite distinct. Further, different events called for different styles, as did the status and social requirements for those who attended. For the first time in the clothing history of the culturally elite, different distinct fashions overlapped rather than replacing each other, the clothing choices marking divisions in this class.

The century saw Payne's &quot;Six basic forms&quot; or silhouettes generally in this order but sometimes overlapping:

# '''Fullness in the back'''. The fabric bustle. While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says.&lt;ref name=&quot;:11&quot; /&gt;{{rp|411}} The overskirt was all pulled to the back, the fullness probably mostly made by bunched fabric.
# '''The round skirt'''. &quot;The bell or dome shape resulted from the reintroduction of hoops[,] in England by 1710, in France by 1720.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|411}}
# '''The ellipse, panniers'''. &quot;The ellipse ... was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. ... English court costume [411/413] followed this fashion well into the nineteenth century.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|411, 413}}
# '''Fullness in the back and sides'''. &quot;The dairy maid, or [[Social Victorians/Terminology#Polonaise|polonaise]], style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise .... These diversions appeared in the late [seventeen] sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}}
# '''Fullness in the back'''. The return of the bustle in the 1780s.&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}}
# '''No fullness'''. The tubular [or Empire] form, drawn from classic art, in the 1790s.&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}}
Hoops affected how women sat, went through doors and got into carriages, as well as what was involved in the popular dances. Length of skirts and trains. Some doorways required that women wearing wide panniers turn sideways, which undermined the &quot;entrance&quot; they were expected to make when they arrived at an event. Also, a woman might be accompanied by a gentleman, who would also be affected by her panniers and the width of the doorway. Over the century skirts varied from ankle length to resting on the floor. Women wearing panniers would not have been able to stand around naturally: the panniers alone meant they had to keep their elbows bent. 

[[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Wooden and Fabric-covered Structure for 18th-century Panniers|left]][[File:Hoop petticoat and corset England 1750-1780 LACMA.jpg|thumb|Hooped Petticoat and Corset, 1750–80]]The 1760–1770 French panniers (left) are &quot;a rare surviving example&quot;&lt;ref name=&quot;:15&quot;&gt;{{Citation|title=Panniers|url=https://www.metmuseum.org/art/collection/search/139668|date=1760–70|accessdate=2025-01-01}}. The Costume Institute, Metropolitan Museum of Art. https://www.metmuseum.org/art/collection/search/139668.&lt;/ref&gt; of the structure of this foundation garment. Almost no examples of panniers survive. The hoops are made with bent cane, held together with red velvet silk ribbon that looks pinked. The cane also appears to be covered with red velvet, and the hoops have metal &quot;hinges that allow [them] to be lifted, facilitating movement in tight spaces.&quot;&lt;ref name=&quot;:15&quot; /&gt; This inventive hingeing permitted the wearer to lift the bottom cane and her skirts, folding them up like an accordion, lifting the front slightly and greatly reducing the width (and making it easier to get through doors). ['''Write the Met to ask about this description once it's finished. Are there examples of boned or metal panniers that they're aware of?''']

The corset and hoops shown (right) are also not reproductions and are also rare examples of foundation garments surviving from the 18th century. These hoops are made with cane held in place by casings sewn into a plain-woven linen skirt.&lt;ref&gt;{{Cite web|url=https://collections.lacma.org/node/214714|title=Woman's Hoop Petticoat (Pannier) {{!}} LACMA Collections|website=collections.lacma.org|access-date=2025-01-03}} Los Angeles County Museum of Art. https://collections.lacma.org/node/214714.&lt;/ref&gt; These 1750–1780 hoops are modestly wide, but the gathering around the casings for the hoops suggests that the panniers could be widened if longer hoops were inserted. (The corset shown with these hoops is treated in the [[Social Victorians/Terminology#Corsets|Corsets section]]. The mannequin is wearing a [[Social Victorians/Terminology#Chemise|chemise undergarment]] as well.)[[File:Johanna Gabriele of Habsburg Lorraine1 copy.jpg|thumb|Martin van Meytens, Johanna Gabriele of Habsburg Lorraine, c. 1760|left]]In her c. 1760 portrait (left), Johanna Gabriele of Habsburg Lorraine is wearing exaggerated court-dress panniers, shown here about the widest that they got. Johanna Gabriele was the daughter of Maria Theresa of Austria, so she was a sister of Marie Antoinette, who also would have worn panniers as exaggerated as these. Johanna Gabriele's hairstyle has not grown into the huge bouffant style that developed to balance the wide court dress, so her outfit looks out of proportion in this portrait. And, because of her panniers, her arms look slightly awkward. The tips of her shoes show because her skirt has been pulled back and up to rest on them.

France had become the leader in high fashion by the middle of the century, led first by Madame Pompadour and then by Marie Antoinette, who was crowned queen in 1774.&lt;ref&gt;{{Cite journal|date=2025-04-23|title=Marie Antoinette|url=https://en.wikipedia.org/wiki/Marie_Antoinette|journal=Wikipedia|language=en}}&lt;/ref&gt;  Court dress has always been regulated, but it could be influenced. Marie Antoinette's influence was toward exaggeration, both in formality and in informality. In their evolution formal-dress skirts moved away from the body in front and back but were still wider on the sides and were decorated with massive amounts of trim, including ruffles, flowers, lace and ribbons. The French queen led court fashion into greater and greater excess: &quot;Since her taste ran to dancing, theatrical, and masked escapades, her costumes and those of her court exhibited quixotic tendencies toward absurdity and exaggeration.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|428}} Both Madame Pompadour's and Marie Antoinette's taste ran to extravagance and excess, visually represented in the French court by the clothing.[[File:Marie Antoinette 1778-1783.jpg|thumb|Marie Antoinette in 1778 and 1779]]The two portraits (right), painted by Élizabeth Louise Vigée Le Brun in 1778 on the left and 1779 on the right, show Marie Antoinette wearing the same dress. Although one painting has been photographed as lighter than the other, the most important differences between the two portraits are slight variations in the pose and the hairstyle and headdress. Her hair in the 1779 painting is in better proportion to her dress than it is in the earlier one, and the later headdress — a stylized mobcap — is more elaborate and less dependent on piled-up hair. (The description of the painting in Wikimedia Commons says she gave birth between these two portraits, which in particular affected her hair and hairline.&lt;ref&gt;&quot;File:Marie Antoinette 1778-1783.jpg.&quot; ''Wikimedia Commons'' [&lt;bdi&gt;Élisabeth Louise Vigée Le Brun, 2 portraits of Marie Antoinette&lt;/bdi&gt;] https://commons.wikimedia.org/wiki/File:Marie_Antoinette_1778-1783.jpg.&lt;/ref&gt;)[[File:Queen Charlotte, by studio of Thomas Gainsborough.jpg|thumb|Queen Charlotte of England, 1781|left]]

In this 1781&lt;ref&gt;{{Cite web|url=https://artsandculture.google.com/asset/wd/jAGip1dpEkf-Fw|title=Portrait of Queen Charlotte of England - Thomas Gainsborough, studio|website=Google Arts &amp; Culture|language=en|access-date=2025-04-16}}&lt;/ref&gt; portrait from the workshop of Thomas Gainsborough (left), Queen Charlotte is wearing panniers less exaggerated in width than Johanna Gabriele's. The English did not usually wear panniers as wide as those in French court dress, but the decoration and trim on the English Queen Charlotte's gown are as elaborate as anything the French would do.

The ruffles (many of them double) and fichu are made with a sheer silk or cotton, which was translucent rather than transparent. The ruffles on Queen Charlotte's sleeves are made of lace. The ruffles and poufs of sheer silk are edged in gold. The embroidered flowers and stripes, as well as the sequin discs and attached clusters are all gold. The skirt rose above the floor, revealing Queen Charlotte's pointed shoe. Shoes were fashion accessories because of the shorter length of the skirts.

The whole look is more balanced because of the bouffant hairstyle, the less extreme width in the panniers and the greater fullness in front (and, probably, back). 

The white dress worn by the queen in Season 1, Episode 4 of the BBC and Canal+ series ''Marie Antoinette'' stands out because nobody else is wearing white at the ball in Paris and because of the translucent silk or muslin fabric, which would have been imported from India at that time (some silk was still being imported from China). Muslin is not a rich or exotic fabric to us, but toward the end of the 18th century, muslin could be imported only from India, making it unusual and expensive.&lt;blockquote&gt;Another English contribution to the fashion of the eighties was the sheer white muslin dress familiar to us from the paintings of Reynolds, Romney, and Lawrence. In this respect the English fell under the spell of classic Greek influence sooner than the French did. Lacking the restrictions imposed by Marie Antoinette's court, the English were free to adapt costume designs from the source which was inspiring their architects and draftsmen.&lt;ref name=&quot;:11&quot; /&gt;{{rp|438}} &lt;/blockquote&gt;So while a sheer white dress would have been unlikely in Marie Antoinette's court, according to Payne, the fabric itself was available and suddenly became very popular, in part because of its simplicity and its sheerness. The Empire style replaced the Rococo busyness in a stroke, like the French Revolution.

By the 1790s French and English fashion had evolved in very different directions, and also by this time, accepted fashion and court dress had diverged, with the formulaic properties of court dress — especially in France — preventing its development. In general,&lt;blockquote&gt;English women were modestly covered ..., often in overdress and petticoat; that heavier fabrics with more pattern and color were used; and that for a while hairdress remained more elaborate and headdress more involved than in France.&lt;ref name=&quot;:11&quot; /&gt;{{rp|441}}&lt;/blockquote&gt;Even in such a rich and colorful court dress as Queen Charlotte is wearing in the Gainsborough-workshop portrait, her more &quot;modest&quot; dress shows these trends very clearly: the white (muslin or silk) and the elaborate style in headdress and hair.

=== Polonaise ===

==== Marie Antoinette — The Context ====

The robe à la Polonaise in casual court dress was popularized by Marie Antoinette for less formal settings and events, a style that occurred at the same time as highly formal dresses with panniers. An informal fashion not based on court dress, although court style would require panniers, though not always the extremely wide ones, and the new style. It was so popular that it evolved into one way court dress could be.[[File:Marie Antoinette in a Park Met DP-18368-001.jpg|thumb|Le Brun, ''Marie Antoinette in a Park'']]Trianon: Marie Antoinette's &quot;personal&quot; palace at Versailles, where she went to entertain her friends in a casual environment. While there, in extended, several-day parties, she and her friends played games, did amateur theatricals, wore costumes, like the stylization of what a dairy maid would wear. A release from the very rigid court procedures and social structures and practices. Separate from court and so not documented in the same way events at Versailles were.

In the c. 1780–81 sketch (right) of Marie Antoinette in a Park by Elisabeth Louise Vigée Le Brun,&lt;ref&gt;Le Brun, Elisabeth Louise Vigée. ''Marie Antoinette in a Park'' (c. 1780–81). The Metropolitan Museum of Art https://www.metmuseum.org/art/collection/search/824771.&lt;/ref&gt; the queen is wearing a robe à la Polonaise with an apron in front, so we see her in a relatively informal pose and outfit. The underskirt, which is in part at least made of a sheer fabric, shows beneath the overskirt and the apron. This is a late Polonaise, more decoration, additions of ribbons, lace, lace, [[Social Victorians/Terminology#Plastics|plastics]], ruffles, which did not exist on actual milkmaid dresses or earlier versions of the robe à la Polonaise. Even though this is a sketch, we can see that this dress would be more comfortable and convenient for movement because the bodice is not boned, and wrinkles in the bodice suggest that she is not likely wearing a corset. 

==== Definition of Terms ====

The Polonaise was a late-Georgian or late-18th-century style, the usage of the word in written English dating from 1773 although ''Polonaise'' is French for ''the Polish woman'', and the style arose in France:&lt;blockquote&gt;A woman's dress consisting of a tight, unboned bodice and a skirt open from the waist downwards to reveal a decorative underskirt. Now historical.&lt;ref name=&quot;:13&quot;&gt;“Polonaise, N. &amp; Adj.” ''Oxford English Dictionary'', Oxford UP, September 2024, https://doi.org/10.1093/OED/2555138986.&lt;/ref&gt;&lt;/blockquote&gt;The lack of boning in the bodice would make this fashion more comfortable than the formal foundation garments worn in court dress.

The term ''á la polonaise'' itself is not in common use by the French nowadays, and the French ''Wikipédia'' doesn't use it for clothing. French fashion drawings and prints from the 18th-century, however, do use the term.

Elizabeth Lewandowski dates the Polonaise style from about 1750 to about 1790,&lt;ref name=&quot;:7&quot; /&gt;{{rp|123}} and Payne says it was &quot;prevalent&quot; in the 1770s.&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}}

The style à la Polonaise was based on an idealization of what dairy maids wore, adapted by aristocratic women and frou-froued up. Two dairymaids are shown below, the first is a caricature of a stereotypical milkmaid and the second is one of Marie Antoinette's ladies in waiting costumed as a milkmaid.
[[File:La laitiere. G.16931.jpg|left|thumb|Mixelle, ''La Laitiere'' (the Milkmaid)]]
[[File:Madame A. Aughié, Friend of Queen Marie Antoinette, as a Dairymaid in the Royal Dairy at Trianon - Nationalmuseum - 21931.tif|thumb|Madame A. Aughié, as a Dairymaid in the Royal Dairy at Trianon]]In the aquatint engraving of ''La Laitiere'' (left) by Jean-Marie Mixelle (1758–1839),&lt;ref&gt;Mixelle, Jean-Marie. ''La Laitiere'', Musée Carnavalet, Histoire de Paris, Inventory Number: G.16931. https://www.parismuseescollections.paris.fr/fr/musee-carnavalet/oeuvres/la-laitiere-8#infos-secondaires-detail.&lt;/ref&gt; the milkmaid is portrayed as flirtatious and, perhaps, not virtuous. She is wearing clogs and two white aprons. Her bodice is laced in front, the ruffle is probably her chemise showing at her neckline, and the peplum sticks out, drawing attention to her hips. As apparently was typical, she is wearing a red skirt, short enough for her ankles to show. The piece around her neck has become untucked from her bodice, contributing to the sexualizing, as does the object hanging from her left hand and directing the eye to her bosom. (The collection of engravings that contains this one is undated but probably from the late 19th or early 20th century.)

The 1787 &lt;bdi&gt;Adolf Ulrik Wertmüller&lt;/bdi&gt; portrait of Madame Adélaïde Aughié in the Royal Dairy at Petit Trianon-Le Hameau&lt;ref&gt;Wertmüller, Adolf Ulrik. ''Adélaïde Auguié as a Dairy-Maid in the Royal Dairy at Trianon''. 1787. The National Museum of Sweden, Inventory number NM 4881. https://collection.nationalmuseum.se/en/collection/item/21931/.&lt;/ref&gt; (right) is about as casual as Le Trianon got. A contemporary of Marie Antoinette, she is in costume as a milkmaid in the Royal Dairy at Trianon, perhaps for a theatrical event or a game. Her dress is not in the à la Polonaise style but a court interpretation of what a milkmaid would look like, in keeping with the hired workers at le Trianon.

==== The 3 Poufs ====
Visually, the style à la Polonaise is defined by the 3 poufs made by the gathering-up of the overskirt. Initially most of the fabric was bunched to make the poufs, but eventually they were padded or even supported by panniers. Payne describes how the polonaise skirt was constructed, mentioning only bunched fabric and not padding:&lt;blockquote&gt;The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, [or, later, buckles] which were used to form the three great ‘poufs’ known as the polonaise .... These diversions [the poufs] appeared in the late [seventeen] sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period.&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}}&lt;/blockquote&gt;[[File:Robe à la polonaise jaune et violette, Galerie des modes, Fonds d'estampes du XVIIIème siècle, G.4555.jpg|thumb|Robe à la polonaise, c. 1775]]The overskirt, which was gathered or pulled into the 3 distinctive poufs, was sometimes quite elaborately decorated, revealing the place of this garment in high fashion (rather than what an actual working dairy maid might wear). The fabrics in the underskirt and overskirt sometimes were different and contrasting; in simpler styles, the two skirts might have the same fabrics. More complexly styled dresses were heavily decorated with ruffles, bows, [[Social Victorians/Terminology#Plastics|plastics]], ribbons, flowers, lace and trim.

The c. 1775&lt;ref name=&quot;:21&quot;&gt;&quot;Robe à la polonaise jaune et violette, Galerie des modes, Fonds d'estampes du XVIIIème siècle.&quot; Palais Galliera, musée de la Mode de la Ville de Paris. Inventory number: G.4555. https://www.parismuseescollections.paris.fr/fr/palais-galliera/oeuvres/robe-a-la-polonaise-jaune-et-violette-galerie-des-modes-fonds-d-estampes-du#infos-principales.&lt;/ref&gt; fashion color print (right) shows the way the overskirt of the Polonaise was gathered into 3 poufs, one in back and one on either side. In this illustration, the underskirt and the overskirt have the same yellow fabric trimmed with a flat band of purple fabric. The 18th-century caption printed below the image identifies it as a &quot;Jeune Dame en robe à la Polonoise de taffetas garnie a plat de bandes d'une autre couleur: elle est coeffée d'un mouchoir a bordures découpées, ajusté avec gout et bordé de fleurs [Young Lady in a Polonaise dress of taffeta trimmed flat with bands of another color: she is wearing a handkerchief with cut edges, tastefully adjusted and bordered with flowers].&quot;&lt;ref name=&quot;:21&quot; /&gt;

The skirt's few embellishments are the tasseled bows creating the poufs. The gathered underskirt falls straight from the padded hips to a few inches above the floor. Her cap is interesting, perhaps a forerunner of the mob cap (here a handkerchief worn as a cap [&quot;mouchoir a bordures découpées&quot;]).

===== The Evolution of the Polonaise into Court Dress =====
Part of the original attraction of the robe à la Polonaise was that women did not wear their usual heavy corsets and hoops, which is what would have made this style informal, playful, easy to move in, an escape from the stiffness of court life. Traditionally court dress with panniers and the robe à la Polonaise were thought to be separate, competing styles, but actually the two styles influenced each other and evolved into a design that combined elements from both.

By the time the robe à la Polonaise became court dress, the poufs were no longer only bunched fabric but large, controlled elaborations that were supported by structural elements, and the silhouette of the dress had returned to the ellipsis shape provided by panniers, with perhaps a little more fullness in front and back. The underskirt fell straight down from the hip level, indicating that some kind of padding or structure pulled it away from the body.

Court dress required the controlled shape of the skirt and a tightly structured bodice, which could have been achieved with corseting or tight lacing of the bodice itself. In the combined style, the bodice comes to a pointed V below the waist, which could only be kept flat by stays. While the Polonaise was ankle length, court dress touched the floor.

The following 3 images are fashion prints showing Marie Antoinette in court dress influenced by the robe à la Polonaise, made into a personal style for the queen by the asymmetrical poufs, the reduction of Rococo decoration, layers stacked upon each other and a length that keeps the hem of the skirts off the floor.[[File:Marie Antoinette de modekoningin Gallerie des Modes et Costumes Français Gallerie des Modes et Costumes Français, 1787, ooo 356 Grand habit de bal a la Cour (..), RP-P-2009-1213.jpg|thumb|Marie Antoinette in a Court Ball Gown à la Polonaise|left]]The 1787 &quot;Grand habit de bal à la Cour, avec des manches à la Gabrielle &amp; c.&quot; (left) by printmaker Nicolas Dupin, after a drawing by Augustin de Saint-Aubin, shows Marie Antoinette in a ballgown for the court with sleeves à la Gabrielle.&lt;ref&gt;{{Cite web|url=https://www.rijksmuseum.nl/en/collection/object/Marie-Antoinette-The-Queen-of-Fashion-Gallerie-des-Modes-et-Costumes-Francais--10ceb0e05fbb45ad4941bed1dacb27f1|title=Marie Antoinette: The Queen of Fashion: Gallerie des Modes et Costumes Français|website=Rijksmuseum.nl|language=en|access-date=2025-05-02}}&lt;/ref&gt; 

This ballgown, influenced by the robe à la polonaise, is balanced but asymmetrical and seems to have panniers for support of the side poufs. The only decoration on the skirt is ribbon or braid and tassels. Contrasting fabrics replace the [[Social Victorians/Terminology#Frou-frou|frou-frou]] for more depth and interest. The lining of the poufs has been pulled out for another contrasting color. The print makes it impossible to tell if the purple is an underskirt and an overskirt or one skirt with attached loops of the ribbon-like trim. 

(A sleeve à la Gabrielle has turned out to be difficult to define. The best we can do, which is not perfect, is a 4 July 1814 description: &quot;On fait, depuis quelque temps, des manches à la Gabrielle. Ces manches, plus courtes que les manches ordinaires, se terminent par plusieurs rangs de garnitures. Au lieu d'un seul bouillonné au poignet, on en met trois ou quatre, que l'on sépare par un poignet.&quot;&lt;ref&gt;&quot;Modes.&quot; ''Journal des Dames et des Modes''. 4 July 1814 (18:37), vol. 10, 1. ''Google Books'' https://books.google.com/books?id=kwNdAAAAcAAJ.&lt;/ref&gt;{{rp|296}} [&quot;For some time now, sleeves have been made in the Gabrielle style. These sleeves, shorter than ordinary sleeves, end in several rows of trimmings. Instead of a single ruffle at the wrist, three or four are used, separated by a wrist treatment.&quot;] The sleeves on the bodice of robes à la Polonaise seem to have been short, 3/4-length or less.) [[File:Gallerie des Modes et Costumes Français, 1787, sss 384 Robe de Cour à la Turque (..), RP-P-2009-1220.jpg|thumb|Marie Antoinette in a Court Dress à la Turque]]The c. 1787 &quot;Robe de Cour à la Turque, coeffure Orientale aves des aigrettes et plumes, &amp;c.&quot; (right) by printmaker Nicolas Dupin, after a drawing by Augustin de Saint-Aubin, shows Marie Antoinette in a court dress à la Turque with a headdress that has [[Social Victorians/Terminology#Aigrette|aigrettes]] and plumes.&lt;ref&gt;{{Cite web|url=https://www.rijksmuseum.nl/en/collection/object/---75499afec371ac1741dd98d769b14698|title=Gallerie des Modes et Costumes Français, 1787, sss 384 : Robe de Cour à la Turque; (...)|website=Rijksmuseum.nl|language=en|access-date=2025-05-02}}&lt;/ref&gt; The &quot;coeffure Orientale&quot; seems to be a highly stylized turban.

This court dress is à la Polonaise in that it has poufs, but it has 2 layers of poufs and an underskirt with a large ruffle. With its unusual striped fabric, its contrasting colors, the very asymmetrical skirt and the ruffles, bows and tassels, this is an elaborate and visually complex dress, but it is not decorated with a lot of [[Social Victorians/Terminology#Frou-frou|frou-frou]].

Several prints in this fashion collection show the robe à la Turque, a late-Georgian style [1750–1790],&lt;ref name=&quot;:7&quot; /&gt;{{rp|250}} none of which look &quot;Turkish&quot; in the slightest. Lewandowski defines robe à la Turque:&lt;blockquote&gt;

Very tight bodice with trained over-robe with funnel sleeves and a collar. Worn with a draped sash.&lt;ref name=&quot;:7&quot; /&gt;{{rp|250}}&lt;/blockquote&gt;

Her &quot;Robe à la Reine&quot; might offer a better description of this outfit, or at least of the overskirt:&lt;blockquote&gt;Popular from 1776 to 1787, bodice with an attached overskirt swagged back to show the underskirt. .... Gown was short sleeved and elaborately decorated.&lt;ref name=&quot;:7&quot; /&gt;{{rp|250}}&lt;/blockquote&gt;[[File:Marie Antoinette de modekoningin Gallerie des Modes et Costumes Français Gallerie des Modes et Costumes Francais, 1787, ooo.359, Habit de Cour en hyver (titel op object), RP-P-2004-1142.jpg|thumb|Marie Antoinette in Winter Court Fashion]]

This 18th-century interpretation of what looked Turkish would have been about what was fashionable and, in the case of Marie Antoinette's court, dramatic.

The 1787 &quot;Habit de Cour en hyver garni de fourrures &amp;c.&quot; (right) of Marie Antoinette by printmaker Nicolas Dupin, after a drawing by Augustin de Saint-Aubin, shows Marie Antoinette in a winter court outfit trimmed with white fur.&lt;ref&gt;{{Cite web|url=https://www.rijksmuseum.nl/en/collection/object/Marie-Antoinette-The-Queen-of-Fashion-Gallerie-des-Modes-et-Costumes-Francais--727dc366885cc0596cd60d7b2c57e207|title=Marie Antoinette: The Queen of Fashion: Gallerie des Modes et Costumes Français|website=Rijksmuseum.nl|language=en|access-date=2025-05-02}}&lt;/ref&gt; Unusually, this &quot;habit&quot; à la Polonaise has a train. The highly stylized court version of a mob cap was appropriated from the peasantry and turned into this extravagant headdress with its unrealistic high crown and its huge ribbon and bows. This outfit as a whole is balanced even though individual elements (like the cap and the white drapes gathered and bunched with bows and tassels) are out of proportion.

The decadence of the aristocratic and royal classes in France at the end of the 18th century are revealed by these extravagant, dramatic fashions in court dress. These restructured, redesigned court dresses are the merging of the earlier, highly decorated and formal pannier style with the simpler, informal style à la Polonaise. The design is complex, but the complexity does not result from the variety of decorations. The most important differences in the merged design are in the radical reduction of frou-frou and the number of layers. Also, sometimes, the skirts are ankle rather than floor length. The foundation garments held the layers away from the legs, not restricting movement. The different styles of farthingales that existed at the same time are variations on a theme, but the panniers and the Polonaise styles, which also existed at the same time, had different purposes and were designed for different events, but the two styles influenced each other to the point that they merged. 

All the various forms of hoops we've discussed so far are not discrete but moments in a long evolution of foundation structures. Once fashion had moved on, they all passed out of style and were not repeated. Except the Polonaise, which had influence beyond the 18th century — in the 1870s revival of the à la Polonaise style and in Victorian fancy-dress (or costume) balls. For example, [[Social Victorians/People/Pembroke#Lady Beatrix Herbert|Lady Beatrix Herbert]] at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] was wearing a Polonaise, based on a Thomas Gainsborough portrait of dancer Giovanna Baccelli.

=== Crinoline Hoops ===
''[[Social Victorians/Terminology#Crinoline|Crinoline]]'', technically, is the name for a kind of stiff fabric made mostly from horsehair and sometimes linen, stiffened with starch or glue, and used for [[Social Victorians/Terminology#Foundation Garments|foundation garments]] like petticoats or bustles. The term ''crinoline'' was not used at first for the cage (shown in the image below left), but that kind of structure came to be called a crinoline as well as a cage, and the term is still used in this way by some.

After the 1789 French Revolution, for about one generation, women stopped wearing corsets and hoops in western Europe.&lt;ref name=&quot;:11&quot; /&gt;{{rp|445–446}} What they did wear was the Empire dress, a simple, columnar style of light-weight cotton fabric that idealized classical Greek outlines and aesthetics. Cotton was a fabric for the elite at this point since it was imported from India or the United States. Sometimes women moistened the fabric to reveal their &quot;natural&quot; bodies, showing that they were not wearing artificial understructures.[[File:Crinoline era3.gif|thumb|1860s Cage Showing the Structure|left]]

Beginning in the second decade of the 19th century and continuing through the 1830s, corsets returned and skirts became more substantial, widened by layers of flounced cotton petticoats — and in winter, heavy woolen or quilted ones. The waist moved down to the natural waist from the Empire height. As skirts got wider in the 1840s, the petticoats became too bulky and heavy, hanging against the legs and impeding movement. In the mid 1850s&lt;ref name=&quot;:11&quot; /&gt;{{rp|510}} &lt;ref name=&quot;:7&quot; /&gt;{{rp|78}} those layers of petticoats began to be replaced by hoops, which were lighter than all that fabric, even when made of steel, and even when really wide.

Lewandowski defines 3 kinds of 19th-century cages:&lt;blockquote&gt;cage: Crinoline (1840–1865 C.E.) to Bustle (1865–1890 C.E.). United Kingdom. Nickname for artificial crinoline; petticoat with whalebone hoops, wire, or watch-string.

cage Americaine: Crinoline (1840–1865 C.E.). France. Petticoat in which only bottom half was covered with fabric, upper half only boning.

cage empire: Crinoline (1840–1865 C.E.) to Bustle (1865–1890 C.E.). Popular from 1861 to 1869, slightly trained petticoat made of 30 steel hoops that increased in size as they approached the ground.&lt;ref name=&quot;:7&quot; /&gt; (46)&lt;/blockquote&gt;

R. C. Milliett patented the first cage, or crinoline hoops in 1856 in Paris,&lt;ref&gt;&quot;The Fashion.&quot; Citing the Collection of the Kent State University Museum. ''Facebook'' 6 August 2025. https://www.facebook.com/photo/?fbid=122200374008095594&amp;set=a.122128150262095594. The Fashion's WhatsApp channel: 

https://whatsapp.com/channel/0029VbBPfXc2UPBIy6Aj651n.&lt;/ref&gt; but cages were in use before the patent. Empress Eugénie of France, wife of Napoleon III, used the cage in 1855 to obscure evidence of pregnancy, which let her be more present in public:&lt;blockquote&gt;“On November 23, 1855, Lord Malmesbury went to a dinner at the Tuileries and found Eugénie “looking very handsome, and all appearances concealed by the large dresses now worn.”&lt;ref name=&quot;:22&quot;&gt;Goldstone, Nancy. ''The Rebel Empresses: Elisabeth of Austria and Eugénie of France, Power and Glamour in the Struggle for Europe''. Little, Brown, 2025.&lt;/ref&gt;{{rp|296}}&lt;/blockquote&gt;

The caged crinoline was Eugénie's&lt;blockquote&gt;signature, over-the-top look. An update on the eighteenth-century pannier worn by her muse, Marie Antoinette, the caged crinoline created a skirt so broad that it often made it difficult for a woman wearing one to get through a doorway [like the court panniers of Marie Antoinette's time]. Because they were all the rage at the French court, crinolines were immensely popular for years — Sisi [Elisabeth, Empress of Austro-Hungary and the Holy Roman Empire as well as Queen Victoria] owned one ... — but for Eugenie, the dome-shaped skirts had the added advantage, as Malmesbury pointed out, of hiding her condition in case she miscarried again.&lt;ref name=&quot;:22&quot; /&gt;{{rp|296, n. vi}}&lt;/blockquote&gt;

The sketch (above left) shows a crinoline cage from the 1850s and 1860s, making clear the structure that underlay the very wide, bell or hemisphere shapes of the era without the fabric that would normally have covered it.&lt;ref&gt;Jensen, Carl Emil. ''Karikatur-album: den evropaeiske karikature-kunst fra de aeldste tider indtil vor dage. Vaesenligst paa grundlag af Eduard Fuchs : Die karikature'', Eduard Fuchs. Vol. 1. København, A. Chrustuabsebs Forlag, 1906. P. 504, Fig. 474 (probably) ''Google Books'' https://books.google.com/books?id=BUlHAQAAMAAJ.&lt;/ref&gt; (This image was published in a book in 1904, but it may have been drawn earlier. The [[Social Victorians/Terminology#Chemise|chemise]] is accurate but oversimplified, minus the usual ruffles, more for the wealthy and less for the working classes.) '''The common underwear of this time would have been two individual legs connected at the waist, at most. The woman's crotch would not be enclosed, leaving her exposed if she fell or the wind was strong enough to lift her skirts far enough.''' 

[[Social Victorians/People/Louisa Montagu Cavendish|Louise, Duchess of Manchester (later Duchess of Devonshire)]] must have been wearing a cage like this in 1859 when one of her hoops caught in a stile she was crossing and she fell. She landed &quot;on her feet with her cage and whole petticoats remaining above her head,&quot; revealing &quot;to all the world in general and the Duc de Malakoff in particular&quot; that she was wearing &quot;a pair of scarlet tartan knickerbockers,&quot; the kind of garment men would wear when hunting.&lt;ref name=&quot;:202&quot;&gt;Vane, Henry. ''Affair of State: A Biography of the 8th Duke and Duchess of Devonshire''. Peter Owen, 2004.&lt;/ref&gt;

When people think of 1860s hoops, they think of this shape, the one shown in, say, the 1939 film ''Gone with the Wind''. The extremely wide, round shape, which is what we are accustomed to seeing in historical fiction and among re-enactors, was very popular in the late 1850s and early 1860s, but it was not the only shape hoops took at this time. The half-sphere shape — in spite of what popular history prepares us to think — was far from universal.[[File:Miss Victoria Stuart-Wortley, later Victoria, Lady Welby (1837-1912) 1859.jpg|thumb|Victoria Stuart-Wortley, 1859]]As the 1860s progressed, hoops (and skirts) moved towards the back, creating more fullness there and leaving a flatter front. The photographs below show the range of choices for women in this decade. Cages could be more or less wide, skirts could be more or less full in back and more or less flat in front, and skirts could be smooth, pleated or folded, or gathered. Skirts could be decorated  with any of the many kinds of ruffles or with layers (sometimes made of contrasting fabrics), and they could be part of an outfit with a long bodice or jacket (sometimes, in fact, a [[Social Victorians/Terminology#Peplum|peplum]]). As always, the woman's social class and sense of style, modesty and practicality affected her choices.

In her portrait (right) Victoria Stuart-Wortley (later Victoria, Lady Welby) is shown in 1859, two years before she became one of Queen Victoria's maids of honor. While Stuart-Wortley is dressed fashionably, her style of clothing is modest and conservative. The wrinkles and folds in the skirt suggest that she could be wearing numerous petticoats (which would have been practical in cold buildings), but the smoothness and roundness of the silhouette of the skirt suggest that she is wearing conservative hoops.[[File:Elisabeth Franziska wearing a crinoline and feathered hat.jpg|thumb|Archduchess Elisabeth Franziska, 1860s|left]]

The portrait of Archduchess Elisabeth Franziska (left) offers an example of hoops from the 1860s that are not half-sphere shaped and a skirt that is not made to fit smoothly over them. The dress seems to have a short peplum whose edges do not reach the front. She is standing close to the base of the column and possibly leaning on the balustrade, distorting the shape of the skirt by pushing the hoop forward.

This dress has a complex and sophisticated design, in part because of the weight and textures of the fabric and trim. The folds in the skirt are unusually deep. Even though the textured or flocked fabric is light-colored, this could be a winter dress.

The skirt is trimmed with zig-zag rows of ruffles and a ruffle along the bottom edge. The ruffles may be double with the top ruffle a very narrow one (made of an eyelet or some kind of textured fabric). Both the top and bottom edges of the tiered double ruffles are outlined in a contrasting fabric, perhaps of ribbon or another lace, perhaps even crocheted. Visual interest comes from the three-dimensionality provided by the ruffles and the contrast caused by dark crocheted or ribbon edging on the ruffles. In fact, the ruffles are the focus of this outfit.

[[File:Her Majesty the Queen Victoria.JPG|thumb|Queen Victoria at Windsor Castle, 1861]]
The photographic portrait (right) of Queen Victoria at Windsor Castle, in evening dress with diadem and jewels, is by Charles Clifford&lt;ref&gt;{{Cite web|url=https://wellcomecollection.org/works/ppgcfuck|title=Queen Victoria. Photograph by C. Clifford, 1861.|website=Wellcome Collection|language=en|access-date=2025-02-03}}&lt;/ref&gt; of Madrid, dated 14 November 1861 and now held by the Wellcome Institute. Prince Albert died on 14 December 1861,&lt;ref&gt;{{Cite journal|date=2025-01-20|title=Prince Albert of Saxe-Coburg and Gotha|url=https://en.wikipedia.org/wiki/Prince_Albert_of_Saxe-Coburg_and_Gotha|journal=Wikipedia|language=en}}&lt;/ref&gt; so this carte-de-visite portrait was taken one month before Victoria went into mourning for 40 years.

This fashionable dress could be a ballgown designed by a designer.

The hoops under these skirts appear to be round rather than elliptical but are rather modest in their width and not extreme. That is, there is as much fullness in the front and back as on the sides. In this style, the skirt has a smooth appearance because it is not fuller at the bottom than the waist, where it is tightly gathered or pleated, so the skirts lie smoothly on the hoops and are not much fuller than the hoops. The smoothness of this skirt makes it definitive for its time. 

Instead of elaborate decoration, this visually complex dress depends on the woven moiré fabric with additional texture created by the shine and shadows in the bunched gathering of the fabric. The underskirt is gathered both at the waist and down the front, along what may be ribbons separating the gathers and making small horizontal bunches. The overskirt, which includes a train, has a vertical drape caused by the large folds at the waist. The horizontal design in the moiré fabric contrasts with the vertical and horizontal gathers of the underskirt and large, strongly vertical folds of the overskirt.[[File:Queen Victoria photographed by Mayall.JPG|thumb|Queen Victoria photographed by Mayall. early 1860s|left]]
The carte-de-visite portrait of Queen Victoria by John Jabez Edwin Paisley Mayall (left) shows hoops that are more full in the back than the front. Mayall took a number of photographs of the royal family in 1860 and in 1861 that were published as cartes de visite,&lt;ref&gt;{{Cite journal|date=2024-11-08|title=John Jabez Edwin Mayall|url=https://en.wikipedia.org/wiki/John_Jabez_Edwin_Mayall|journal=Wikipedia|language=en}}&lt;/ref&gt; and the style of Victoria's dress is consistent with the early 1860s.

The fact that she has white or a very light color at her collar and wrists suggests that she was not in full mourning and thus wore this dress before Prince Albert died on 14 December 1861. We cannot tell what color this dress is, and it may not be black in spite of how it appears in this photograph. Victoria's hoops are modest — not too full — and mostly round, slightly flatter in the front. The skirt gathers more as it goes around the sides to the back and falls without folds in the front, where it is smoother, even over the flatter hoops. This is a winter garment with bulky sleeves and possibly fur trim. Except for what may be an undergarment at the wrists, this one-layer garment might be a dress or a bodice and skirt (perhaps with a short jacket). Over-trimmed garments were standard in this period. Lacking layers, ruffles, lace or frou-frou, the simple design of Victoria's dress is deliberate and balanced — and looks warm.

The bourgeois, inexpensive-looking design of this dress echoes Victoria's performance of a queen who is respectable and responsible rather than aristocratic and &quot;fashion forward.&quot; So she looks like a middle-class matron.[[File:Queen Emma of Hawaii, photograph by John &amp; Charles Watkins, The Royal Collection Trust (crop).jpg|thumb|Queen Emma Kaleleokalani of Hawai'i, 1865]]

The portrait (right) of Queen Emma of Hawaii —  Emma Kalanikaumakaʻamano Kaleleonālani Naʻea Rooke — is a carte de visite from an album of ''Royal Portraits'' that Queen Victoria collected. The carte-de-visite photograph is labelled 1865 and ''Queen Emma of the Sandwich Islands'',&lt;ref&gt;Unknown Photographer. ''Emma Kalanikaumakaʻamano Kaleleonālani Naʻea Rooke, Queen of the Kingdom of Hawaii (1836-85)''. ''www.rct.uk''. Retrieved 2025-02-07. https://www.rct.uk/collection/2908295/emma-kalanikaumakaamano-kaleleonalani-naea-rooke-queen-of-the-kingdom-of-hawaii.&lt;/ref&gt; possibly in Victoria's hand. How Victoria got this photograph is not clear. Queen Emma traveled to North America and Europe between 6 May 1865 and 23 October 1866,&lt;ref&gt;Benton, Russell E. ''Emma Naea Rooke (1836-1885), Beloved Queen of Hawaii''. Lewiston, N.Y., U.S.A. : E. Mellen Press, 1988. ''Internet Archive'' https://archive.org/details/emmanaearooke1830005bent/.&lt;/ref&gt;{{rp|49}} visiting London twice, the second time in June 1866.&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|date=2025-01-07|title=Queen Emma of Hawaii|url=https://en.wikipedia.org/wiki/Queen_Emma_of_Hawaii|journal=Wikipedia|language=en}}&lt;/ref&gt;

In her portrait Queen Emma is standing before some books and an open jewelry box. She shows an elegant sense of style.

The silhouette shows a sophisticated variation of the hoops as the fullness has moved to the back and the front flattened. The large pleats suggest a lot of fabric, but the front falls almost straight down. The overskirt and bodice are made from a satin-weave fabric, and the petticoat has a matt woven surface. The overskirt is longer in the back, leading us to expect the petticoat also to be longer and to turn into a train. Although the hoops cause the skirt to fall away from her body in back, the skirt does not drag on the floor as a train would and just clears the floor all the way around.

This optical illusion of a train makes this dress look more formal than it actually was. The covered shoulders and décolletage say the dress was not a formal or evening gown. In fact, this looks like a winter dress, and the sleeves (which she has pushed up above her wrist) are wrinkled, suggesting they may be padded. Queen Emma seems to have worn veils like this at other times as well, especially after the death of her husband, as did Victoria, so this is also not her wedding dress.

Popular history has led us to believe that crinoline hoops were half-spherical and always very wide, but photographs of the time show a variety of shapes for skirts, with many women wearing skirts that had flatter fronts and more fabric in the back. In fact, also in the 1860s, according to Lewandowski, a version of the bustle — called a crinolette or crinolette petticoat — developed:&lt;blockquote&gt;Crinolette petticoat: Bustle (1865–1890 C.E.). Worn in 1870 and revived in 1883, petticoat cut flat in front and with half circle steel hoops in back and flounces on bottom back.&lt;ref name=&quot;:7&quot; /&gt;{{rp|78}}&lt;/blockquote&gt;

This development of a bustle mid century is the result of construction techniques that include foundation structures and specifically shaped pattern pieces to achieve the evolving silhouette, in this case part of the general movement of the fullness of skirts away from the front and toward the back. The other essential element of these construction techniques is angled seams in the skirts, made by gores, pieces of fabric shaped to fit the waist (and sometimes the hips) and to widen at the bottom so that the skirt flares outward.

==== The 19th-century Revival of the Polonaise ====
The Polonaise style was revived in the last third of the 19th century, but the revival did not bring back the 18th-century 3 poufs. The robe à la Polonaise had evolved. The foundation that created the poufs is gone, replaced possibly in fact by the crinolette petticoat or something like it. The panniers — and the 2 side poufs they supported — have gone, and the bulk of the fabric has been bunched in the back.

Also, the poufs on the sides have been replaced with a flat drape in front that functions as an overskirt.

The Polonaise dress (below left and right), in the collection of the Los Angeles County Museum of Art, is English, dating from about 1875.&lt;ref name=&quot;:18&quot;&gt;&quot;Woman's Dress Ensemble.&quot; Costumes and Textiles. LACMA: Los Angeles County Museum of Art. https://collections.lacma.org/node/214459.&lt;/ref&gt; The sheer fabric has red &quot;wool supplementary patterning&quot; woven into the weft.&lt;ref name=&quot;:18&quot; /&gt; Because the mannequin is modern, we cannot be certain how long the skirts would have been on the woman who wore this dress.[[File:Woman's Polonaise Dress LACMA M.2007.211.777a-f (1 of 4).jpg|thumb|English Polonaise, c. 1875, front view|left]][[File:Woman's Polonaise Dress LACMA M.2007.211.777a-f (4 of 4).jpg|thumb|English Polonaise, c. 1875, side view]]The dress has an overskirt that is draped up toward the back and pulled under the top poof. The underskirt gets fuller at the bottom because it is constructed with gores to create the A-line but it is also slightly gathered at the waist.

The vertical element is emphasized by the angled silhouette and the folds caused by the gathering at the waist. The ruffles and lace form horizontal lines in the skirts. The skirts are very busy visually because of pattern in the fabric and the contrasting vertical and horizontal elements as well as the ruffles, some of which are double, and the machine-made lace at the edge of the ruffles. The skirts look three dimensional because of these elements and the layering of the fabric, multiplying the jagged-edged red &quot;supplementary patterning.&quot;

The fabric of the overskirt is cut, gathered and draped so that the poufs in back are full and rounded, but they are also possibly supported by some kind of foundation structure. The lower pouf in back introduces the idea that the fullness in the back is layered, making this element of the Polonaise a kind of precursor to the bustle and continuing what the crinolette petticoat began in the 1860s. This layering of the lower pouf also indicates one way a train might be attached.

Laura Ingalls Wilder wrote about the hoops her fictionalized self wore the century before, unusually, and calls her dress a Polonaise. Although they are common in current historical fiction, descriptions of foundation garments are rare in the writings of the women who wore them or in the literature of the time. In ''These Happy Golden Years'' (1943), Wilder gives a detailed description of the undergarments as well as the foundation garments under her dress, including a bustle, and talks about how they make the Polonaise look on her:&lt;blockquote&gt;
Then carefully over her under-petticoats she put on her hoops. She liked these new hoops. They were the very latest style in the East, and these were the first of the kind that Miss Bell had got. Instead of wires, there were wide tapes across the front, almost to her knees, holding the petticoats so that her dress would lie flat. These tapes held the wire bustle in place at the back, and it was an adjustable bustle. Short lengths of tape were fastened to either end of it; these could be buckled together underneath the bustle to puff it out, either large or small. Or they could be buckled together in front, drawing the bustle down close in back so that a dress rounded smoothly over it. Laura did not like a large bustle, so she buckled the tapes in front.

Then carefully over all she buttoned her best petticoat, and over all the starched petticoats she put on the underskirt of her new dress. It was of brown cambric, fitting smoothly around the top over the bustle, and gored to flare smoothly down over the hoops. At the bottom, just missing the floor, was a twelve-inch-wide flounce of the brown poplin, bound with an inch-wide band of plain brown silk. The poplin was not plain poplin, but striped with an openwork silk stripe.

Then over this underskirt and her starched white corset-cover, Laura put on the polonaise. Its smooth, long sleeves fitted her arms perfectly to the wrists, where a band of the plain silk ended them. The neck was high with a smooth band of the plain silk around the throat. The polonaise fitted tightly and buttoned all down the front with small round buttons covered with the plain brown silk. Below the smooth hips it flared and rippled down and covered the top of the flounce on the underskirt. A band of the plain silk finished the polonaise at the bottom.&lt;ref&gt;Wilder, Laura Ingalls. ''These Happy Golden Years.'' Harper &amp; Row, Publishers, 1943. Pp. 161–163.&lt;/ref&gt;&lt;/blockquote&gt;
When a 20th-century Laura Ingalls Wilder calls her character's late-19th-century dress a polonaise, she is probably referring to the &quot;tight, unboned bodice&quot;&lt;ref name=&quot;:13&quot; /&gt; and perhaps a simple, modest look like the stereotype of a dairy maid. While the bodice was unboned, the fact that she is wearing a corset cover means that she is corseted under it.   
==== Bustle or Tournure ====
As we have seen, bustles were popular from around 1865 to 1890.&lt;ref name=&quot;:7&quot; /&gt;{{rp|296}} The French term ''tournure'' was a euphemism in English for ''bustle''. The article on the tournure in the French ''Wikipédia'' addresses the purpose of the bustle and crinoline:&lt;blockquote&gt;
Crinoline et tournure ont exactement la même fonction déjà recherchée à d'autres époques avec le vertugadin et ses dérivés: soutenir l'ampleur de la jupe, et par là souligner par contraste la finesse de la taille; toute la mode du xixe siècle visant à accentuer les courbes féminines naturelles par le double emploi du corset affinant la taille et d'éléments accentuant la largeur des hanches (crinoline, tournure, drapés bouffants…).&lt;ref&gt;{{Cite journal|date=2023-10-27|title=Tournure|url=https://fr.wikipedia.org/wiki/Tournure|journal=Wikipédia|language=fr}}&lt;/ref&gt;

[Translation by ''Google Translate'': Crinoline and bustle have exactly the same function already sought in other periods with the farthingale and its derivatives: to support the fullness of the skirt, and thereby emphasize by contrast the finesse of the waist; all the fashion of the 19th century aimed at accentuating natural feminine curves by the dual use of the corset refining the waist and elements accentuating the width of the hips (crinoline, bustle, puffy drapes, etc.).]&lt;/blockquote&gt;Hoops' final phase was the development of the bustle, which as early as the 1860s was created by one of several methods: by draping the dress over a crinolette petticoat or some other structure, or by pulling the fabric to the back and bunching it with pleats or gathers. The overskirt so popular with the revival of the Polonaise pulled additional fabric to the back of the skirt, the poufs supported by some substructure, bunched fabric, padding and, often, ruffled petticoats. The bustle, then, is more complex than might be normally be thought and more complex than some of the earlier foundation garments in the evolution of hoops, in part because the silhouette of hoops (and dresses) was changing more rapidly in the last half of the 19th century than ever before.

[[File:La Gazette rose, 16 Mai 1874; robe à tournure.jpg|thumb|&quot;Toilettes de Printemps,&quot; 1874|left]]In fact, fashion trends were moving so fast at this point that the two &quot;bustle periods&quot; were actually only two decades, the 1870s and the 1880s. Bustle fashion was at its height for these two decades, which saw the line of the skirts change radically. As the bustle developed, the 1870s ruffles disappeared, replaced by draping and layering, which made the bustles more complex visually. 

&quot;Toilettes de Printemps&quot; (left), an 1874 French fashion plate, shows two women walking in the country, the one in green wearing an extremely long and impractical train. Both of these have several rows of ruffles beneath the overskirt — a short-lived fashion. The ruffles, which disappear in the 2nd bustle period, create a fullness in the front of the skirt at the bottom. The bodice of both dresses connects to an overskirt, like a jacket. The excess skirt fabric is draped in the back over a foundation structure. 

Plumes makes the hats tall, part of the proportioning with the bustle. The dog at the feet of the woman in the green dress recalls the dogs ubiquitous in earlier portraiture. 

The most common image of the bustle — the extreme form of the 1880s — required a complex foundation structure, one of which was &quot;steel springs placed inside the shirring [gathering] around the back of the petticoat.&quot;&lt;ref name=&quot;:7&quot; /&gt; (296) Many manufacturers were making bustles by this time, offering women a choice on the kinds of materials used in the foundation structures ['''check this''']. [[File:Somm26.jpg|thumb|Henry Somm, 1880s]]The Henry Somm watercolor (right) offers a clear example of how extreme bustles got in the mid 1880s, in the 2nd bustle period. Henry Somm was the pen name that François Clément Sommier (1844–1907) used on his paintings.&lt;ref&gt;{{Cite journal|date=2025-02-01|title=Henry Somm|url=https://fr.wikipedia.org/w/index.php?title=Henry_Somm&amp;oldid=222597815|journal=Wikipédia|language=fr}}&lt;/ref&gt; He was in Paris beginning in the 1860s and so was present for the Civil War of 1870–71 and the rise of Impressionism in that highly political and dangerous context.&lt;ref&gt;Smee, Sebastian. ''Paris in Ruins: Love, War, and the Birth of Impressionism''. W. W. Norton, 2024.&lt;/ref&gt;

Somm's c. 1895&lt;ref&gt;&quot;File:Somm26.jpg.&quot; Henry Somm, &quot;An Elegantly Dressed Woman at a Door (wearing mid-1880s bustled fashions),&quot; c. 1895. June 2025. Wikimedia Commons 

https://commons.wikimedia.org/wiki/File:Somm26.jpg.&lt;/ref&gt; impressionist painting shows an immediate moment — an elegant mid-1880s woman outside a door, her right hand and face animated, as if she is talking to someone standing to our left.

Her skirt is quite narrow and flat in front with yards of fabric draped in poufs over the huge foundation bustle behind. This dress has no ruffles or excessive frills. The narrow sleeves and tall hat, along with the umbrella so tightly folded it looks like a stick, contribute to the lean silhouette. Details of the dress are not present because this painting is impressionistic rather than realistic, showcasing the play of light on the fabric and the elegance of the woman. The square corner of the front overskirt is not realistic draping, perhaps an artifact of the painter working from memory rather than a model.[[File:Elizabeth Alice Austen in June 1888.jpg|thumb|Elizabeth Alice Austen, 1888|left]]

The 1888 photograph of American photographer Elizabeth Alice Austen (left) is also from the 2nd bustle period. The very stylish Austen is wearing a bustle that is large but not as extreme as they got. The design of her dress is sophisticated and complex with the proportions more clearly presented than we see in paintings or fashion plates. Her plumed hat is tall, one of the vertical elements, along with the slim line of the bodice, sleeves and skirt. The overskirt is pulled to Austen's right so that it does not lie flat in front. The overskirt and bustle are made from 3 different fabrics with 3 different patterns. The front drape and bodice are made of a light-colored fabric with a light striped pattern, and the bustle has 2 fabrics, a shiny reflective material with no pattern and a strongly striped section that matches the underskirt. The strongly and horizontally striped fabric in the underskirt contrasts with the vertical line of the outfit itself.

In spite of the very strong contrasts in the stripes and horizontal and vertical elements, Austen's dress has a light touch about it. With the draped overskirt in front and the complex construction of the bustle, Austen's dress makes a delicate reference to the poufs of the [[Social Victorians/Terminology#The 19th-century Revival of the Polonaise|Polonaise revival]]. [[File:Cperrien-fashionplatescan-p-vf 33.jpg|thumb|Fashion plate, mid-1880s]]This mid-1880s fashion plate (right) has caricatures for figures, with the usual minuscule waists and feet, exaggerated height and bustles, and general lack of realism in the details of the dresses. In fact, the drawing obscures what is necessary to understand how they were constructed, but it is useful because of the 3 different ways bustles are working in the illustration.

The little girl's overskirt and sash function as a bustle, independent of whatever foundation garments she may be wearing. The two women's outfits have the characteristic narrow sleeves and tall hats, and the one in white is holding another extremely narrow umbrella as well.

The bustle on the red-and-white dress is draped loosely over the very large foundation structure that was typical of the 1880s. The striking red jagged edges define the draping of the overskirt in front and the ruffles on the sides. These ruffles are unlike the ruffles of the 1870s, which added volume. They are flattened essentially into layers, preventing them from sticking out and providing texture rather than fullness. The front overskirt is very flat and the back overskirt contributes to the bustle.

The front of the bodice on both dresses extends to a point determined by the corset and typical of Victorian shaping. The waist treatment on the green dress visually lengthens the point to an extreme. The front of the green skirt is draped and layered. Tiny pleats peep out from below the skirt on both women's dresses. The child's dress has 3 flat pleated ruffles in front that contrast with the fuller but still controlled folds in the back.

These dresses have strongly vertical lines with contrasting horizontal lines in the bustles and trim.

Conclusion

'''Trains, skirt length, movement, materials, one evolutionary process, natural fabrics, accelerating change in fashion, designers and seamstresses, medium of our illustrations'''

== Jewelry and Stones ==

=== Cabochon ===
This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing &quot;the rough parts&quot; and the parts that are not the right stone.&lt;ref&gt;&quot;cabochon, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.&lt;/ref&gt; A cabochon shape is often flat on one side and oval or round, forming a mound in the setting.

=== Cairngorm ===

=== Ferronnière ===
A revival of a Renaissance fashion for controlling the hair and headdress. Usually made of a filet, often with a single pendant stone in the center of the forehead, although the Victorians' ferronnières were often elaborate and encrusted with jewels.&lt;ref&gt;Boyington, Amy. &quot;Ferronnière.&quot; ''History with Amy'' 5 November 2025.

Website fb.watch/FBMyC7bqde [links to fb.watch not allowed].&lt;/ref&gt;

=== Half-hoop ===
Usually of a ring or bracelet, a precious-metal band with a setting of stones on one side, covering perhaps about 1/3 or 1/2 of the band. Half-hoop jewelry pieces were occasionally given as wedding gifts to the bride.

=== Jet ===

=== ''Orfèvrerie'' ===
Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler.

=== Ribbon Necklace ===
=== Solitaire ===
A solitaire is a ring with a single stone set as the focal point. Solitaire rings were occasionally given as wedding gifts to the bride.

=== Turquoise ===

== Mantle, Cloak, Cape ==
In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments.

=== '''Mantle''' ===
A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous.

== Military ==
Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not.

=== Baldric ===
According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is &quot;A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc.&quot;&lt;ref&gt;&quot;baldric, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.&lt;/ref&gt; This sense has been in existence since c. 1300.

=== Cuirass ===
According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is &quot;A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ....&quot;&lt;ref&gt;&quot;cuirass, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.&lt;/ref&gt;

[[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]]
=== Household Cavalry ===
The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of &quot;the Restoration of the Monarchy in 1660.&quot;&lt;ref name=&quot;:3&quot;&gt;Joll, Christopher. &quot;Tales of the Household Cavalry, No. 1. Roles.&quot; The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.&lt;/ref&gt;{{rp|1}} Regimental Historian Christopher Joll says, &quot;the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|1–2}} The 1st and 2nd Life Guards were formed from &quot;the Troops of Horse and Horse Grenadier Guards ... in 1788.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch.

The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 &quot;and guarding their subsequent execution,&quot; the Household Cavalry contributed to the &quot;the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents&quot; in 1884.&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} In 1887 they &quot;were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} 

==== Grenadier Guards ====
Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing &quot;scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs&quot; as well as items associated with weapons and armor.&lt;ref name=&quot;:14&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 34, Col. 2a}}

Founded in England in 1656 as Foot Guards, this infantry regiment &quot;was granted the 'Grenadier' designation by a Royal Proclamation&quot; at the end of the Napoleonic Wars.&lt;ref&gt;{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&amp;oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.&lt;/ref&gt; They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.&lt;ref&gt;{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&amp;oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.&lt;/ref&gt;

At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.&lt;ref&gt;{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&amp;oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.&lt;/ref&gt;

The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,&lt;blockquote&gt;In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.&lt;ref&gt;{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.&lt;/ref&gt;&lt;/blockquote&gt;The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats.

The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:&lt;blockquote&gt;Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.&lt;ref&gt;{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.&lt;/ref&gt;&lt;/blockquote&gt;

==== Life Guards ====
[[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.&lt;ref name=&quot;:14&quot; /&gt;{{rp|p. 36, Col. 3b}}

At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&amp;oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.&lt;/ref&gt; The 1st Life Guards were on &quot;the extreme right&quot; of a French countercharge and &quot;kept their cohesion and consequently suffered significantly fewer casualties.&quot;&lt;ref name=&quot;:4&quot; /&gt;
[[File:Captain, Royal Horse Guards, Blue, England, 1879, from the Military Series (N224) issued by Kinney Tobacco Company to promote Sweet Caporal Cigarettes MET DPB874122.jpg|alt=Old drawing of a soldier wearing a white cuirass, a pointed helmet, thigh-high boots, carrying a long sword|thumb|Captain, Royal Horse Guards, Blue, 1888, a Kinney Brothers Tobacco Company card]]

==== Royal Horse Guards ====
In 1650 the Regiment of Cuirassiers was &quot;raised by Sir Arthur Haselrig on the orders of Oliver Cromwell.&quot;&lt;ref name=&quot;:26&quot;&gt;{{Cite journal|date=2026-05-13|title=Royal Horse Guards|url=https://en.wikipedia.org/w/index.php?title=Royal_Horse_Guards&amp;oldid=1353961278|journal=Wikipedia|language=en}}&lt;/ref&gt; In 1660 &quot;it became the Earl of Oxford's Regiment .... Based on the colour of their uniform, the regiment was nicknamed 'the Oxford Blues', or simply the 'Blues.' In 1750, it became the Royal Horse Guards Blue.&quot;&lt;ref name=&quot;:26&quot; /&gt;

The Royal Horse Guards Blue were moved to Windsor at the end of the 18th century and &quot;acted as royal bodyguards&quot; to George III, who liked them.&lt;ref name=&quot;:26&quot; /&gt; While pay for the men &quot;stagnated,&quot; requirements continued to rise, so that recruits had to come from wealth.&lt;ref name=&quot;:26&quot; /&gt; Riding and hunting skills were helpful to the recruits, who had to provide their own horses, pay for messes and uniforms, not to mention the position itself.&lt;ref name=&quot;:26&quot; /&gt;

They fought in the Battle of Waterloo, with 44 dead, 50 wounded (of which only 6 died).&lt;ref name=&quot;:26&quot; /&gt; With the Duke of Wellington at their head, they became part of the Household Cavalry in 1820.&lt;ref name=&quot;:26&quot; /&gt; An 1890 illustration shows a member of the Royal Horse Guard (above right) fighting at the Battle of Waterloo.

The Royal Horse Guard Blue fought in the Battle of Balaclava in 1854, fighting with the heavy brigades and thus were more successful than the famous light brigade, though conditions were very difficult.&lt;ref name=&quot;:26&quot; /&gt;

A tobacco card published in 1888 (right) shows a captain in the Royal Horse Guards, Blue, in 1879.

In 1884–85 the Blues took part in the attempt to rescue General Gordon in Khartoum. They were sent to South Africa at the end of the 19th century.&lt;ref name=&quot;:26&quot; /&gt;

For those men who were in the Royal Horse Guards at the end of the 19th century, the field marshals were

* 1869–1885: Field Marshal Hugh Rose, 1st Baron Strathnairn, during which time — in 1877 — the name changed to the Royal Horse Guards (The Blues).&quot;&lt;ref name=&quot;:26&quot; /&gt;
* 1885–1895: Field Marshal Sir Patrick Grant
* 1895–1907: Field Marshal Garnet Wolseley, 1st Viscount Wolseley
In 1847 Edmund Packe published his ''[[iarchive:historicalrecord00packiala/|Historical Record of the Royal Regiment of Horse Guards, or Oxford Blues]]'', which has colored images to illustrate the development of the uniform up to the middle of the 19th century (the link goes to the ''Internet Archive'').

== [[Social Victorians/Mourning|Mourning]] ==
== Peplum ==
According to the French ''Wiktionnaire'', a peplum is a &quot;Short skirt or flared flounce layered at the waist of a jacket, blouse or dress&quot; [translation by Google Translate].&lt;ref&gt;{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&amp;oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.&lt;/ref&gt; The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:&lt;blockquote&gt;''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.&lt;ref name=&quot;:5&quot;&gt;“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, &lt;https://doi.org/10.1093/OED/1832614702&gt;.&lt;/ref&gt;&lt;/blockquote&gt;Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.&lt;ref name=&quot;:5&quot; /&gt;

== Revers ==
According to the ''Oxford English Dictionary'', ''revers'' are the &quot;edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge.&quot;&lt;ref&gt;&quot;revers, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.&lt;/ref&gt; The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').&lt;ref&gt;{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&amp;oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.&lt;/ref&gt;

== Traditional vs Progressive Style ==

=== Progressive Style ===
The terms ''artistic dress'' and ''aesthetic dress'' — as well as ''rational dress'' or ''dress reform'' — are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements, as is ''rational dress'', a movement located largely among women in the middle classes from the middle to the end of the century. In general, what we are calling the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. This group of styles was more about individual choices and approaches than the consistent vision offered by couturiers like Maison Worth.

* [[Social Victorians/People/Dressmakers and Costumiers#Alice Comyns Carr and Ada Nettleship|Ada Nettleship]]: Constance Wilde and Ellen Terry; an 1883 exhibition of dress by the Rational Dress Society featured her work, including trousers for women (with a short overskirt)&lt;ref&gt;{{Cite journal|date=2025-04-21|title=Ada Nettleship|url=https://en.wikipedia.org/w/index.php?title=Ada_Nettleship&amp;oldid=1286707541|journal=Wikipedia|language=en}}&lt;/ref&gt;
* [[Social Victorians/People/Dressmakers and Costumiers#Alice Comyns Carr and Ada Nettleship|Alice Comyns Carr]]&lt;ref&gt;{{Cite journal|date=2025-06-06|title=Alice Comyns Carr|url=https://en.wikipedia.org/w/index.php?title=Alice_Comyns_Carr&amp;oldid=1294283929|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Grosvenor Gallery

=== Traditional Style ===
[[File:Victoria Hesse NPG 95941 crop.jpg|alt=Old photograph of a white woman wearing a very tight and fitted bodice with her skirts swept to the back|thumb|Princess Victoria, Marchioness of Milford-Haven (1863–1950), Granddaughter of Queen Victoria; wife of Prince Louis of Battenberg, 1st Marquess, c. 1878]]
Images

* Smooth bodice, fabric draped to the back or covering a bustle with a small cage beneath it:
By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Queen Victoria's granddaughter Princess Victoria is shown (right) wearing a traditional but very stylish c. 1878 dress like one designed by Maison Worth. Blanche Payne describes an example of the 1895 &quot;high style&quot; in a gown by Worth with &quot;the idiosyncrasies of the [1890s] full blown&quot;:&lt;blockquote&gt;The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.&lt;ref name=&quot;:11&quot; /&gt;{{rp|530}}&lt;/blockquote&gt;

The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''.

William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style.

=== The Styles ===
[[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']]
We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — &quot;recording for posterity the aesthetic craze as regards dress&quot; — deliberately contrasts what he calls the &quot;folly&quot; of the Aesthetic Dress movement and the look of the traditional corseted waist.&lt;ref&gt;Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.&lt;/ref&gt; Frith considered the Aesthetic Movement and Aesthetic Dress &quot;ephemeral,&quot; but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century.

As Frith sees it, his painting critiques the &quot;craze&quot; associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or &quot;bones&quot;) to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope.

A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''&lt;ref&gt;Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. &lt;nowiki&gt;https://theses.gla.ac.uk/3279/&lt;/nowiki&gt;&lt;/ref&gt;] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, &quot;while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past.&quot;&lt;ref&gt;Mitchell, Rebecca N. &quot;The Victorian Fancy Dress Ball, 1870–1900.&quot; ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.&lt;/ref&gt;{{rp|294}}

* historical dress: beautiful clothing.
* the range at the ball, from Minnie Paget to Gwladys
* &quot;In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ...&quot;
* The costume of the Duchess of Devonshire does not have a defined waist and may suggest that she herself is not corseted, although that would be a notable departure for her.

Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] 

But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. 

As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it.  

Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. &quot;Every cap, bow, streamer, ruffle, fringe, bustle, glove,&quot; that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls &quot;elaborations&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]].  

Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change.

== Undergarments ==
Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural.
The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. 

=== Undergarments ===

* Chemise
* Corset cover
* Bloomers
* [[Social Victorians/Terminology#Petticoat|Petticoats]] (distinguish between the outer- and undergarment type of petticoat)
* Combinations
* [[Social Victorians/Terminology#Hose, Stockings and Tights|Hose, stockings and tights]]
* Men's shirts
* Men's unders

==== Bloomers ====

==== Chemise ====
A chemise is a garment &quot;linen, homespun, or cotton knee-length garment with [a] square neck&quot; worn under all the other garments except the bloomers or combinations.&lt;ref name=&quot;:7&quot; /&gt; (61) According to Lewandowski, combinations replaced the chemise by 1890.

==== Combinations ====

=== [[Social Victorians/Terminology/Foundation Garments|Foundation Garments]] ===
Foundation structures changed the shape of the body by metal, cane, boning. Men wore corsets as well.
* [[Social Victorians/Terminology#Corset|Corset]]
* [[Social Victorians/Terminology#Hoops|Hoops]]
* Padding

==== Padding ====
Some kinds of padding were used in the Victorian age to enlarge women's bosoms and create cleavage as well as to keep elements of a garment puffy. In the Elizabethan era, men's codpieces are examples of padding.

With respect to the costumes worn at fancy-dress balls, most important would be bum rolls and cod pieces. 

What are commonly called '''bum rolls''' were sometimes called roll farthingales, French farthingales or padded rolls.

== Footnotes ==
{{reflist}}</text>
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    <title>African Arthropods/Chalcidoidea</title>
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      <comment>/* Eulophidae */</comment>
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The megadiverse Superfamily Chalcidoidea has more than 22 000 described species (&gt;2 800 in Africa) and there may be as many as 500 000 undescribed species. Most of these are tiny (less than 3 mm long), and are parasites of a wide variety of insects, spiders and mites, but some chalcidoid larvae eat plant material (in galls or seeds).&lt;ref name=Noyes2003&gt;Noyes, J.S. and Pitkin, B.R. 2003. Universal Chalcidoidea Database: About chalcidoids. The Natural History Museum, London. [https://www.nhm.ac.uk/our-science/data/chalcidoids/introduction.html]&lt;/ref&gt;&lt;ref name=vanNoort2023&gt;van Noort, Simon (2023) WaspWeb: Hymenoptera of the Afrotropical region. www.waspweb.org, accessed on 9 May 2023.&lt;/ref&gt;

[[File:South African chalcidoid wasps in 9 new families 2023.jpg|thumb|600px|South African Chalcidoid wasps moved to new (2018-2022) families: ''Metapelmus'' sp. from Eupelmidae to Metapelmatidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Bohpa maculata'' from Pteromalidae to Ceidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spalangia'' sp. from Pteromalidae to Spalangiidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Spathopus'' sp. from Pteromalidae to Pirenidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Aperilampus'' sp. from Perilampidae to Chrysolampidae
&lt;ref name=Zhang2022&gt;Zhang, J., Heraty, J. M., Darling, C., Kresslein, R. L., Baker, A. J., Torréns, J., Rasplus, J. Y., Lemmon A. &amp; Moriarty Lemmon, E. (2022). Anchored phylogenomics and a revised classification of the planidial larva clade of jewel wasps (Hymenoptera: Chalcidoidea). Systematic Entomology, 47(2), 329-353.  [https://doi.org/10.1111/syen.12533 DOI]&lt;/ref&gt;; ''Megastigmus transvaalensis'' from Torymidae to Megastigmidae&lt;ref name=Janšta2018&gt;Janšta, P., Cruaud, A., Delvare, G., Genson, G., Heraty, J., Křížková, B. and Rasplus, J.Y. (2018). Torymidae (Hymenoptera, Chalcidoidea) revised: molecular phylogeny, circumscription and reclassification of the family with discussion of its biogeography and evolution of life‐history traits. Cladistics, 34(6), pp.627-651. [https://doi.org/10.1111/cla.12228 DOI]&lt;/ref&gt;; ''Lelaps'' or ''Dipara'' sp. from Pteromalidae to Diparidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; ''Lachaisea brevimucro'' from Pteromalidae to Epichrysomallidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;; and ''Solenura nigra'' from Pteromalidae to Lyciscidae&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt;.]]

Before 2018, these species were placed in about 19 accepted families. However, over the past few decades, a host of studies investigating their biology, their morphology, and their DNA have suggested that many more families should be recognized. In 2022 a group of wasp taxonomists agreed to recognize about 50 families worldwide.&lt;ref name=BurksEtAl&gt;{{Cite Q|Q115923766}}&lt;/ref&gt; 

Macro photography has allowed non-scientists a view into the world of these minute creatures, but learning more about them is a challenge. Even with high-quality macro photographs of free-flying wasps, expert hymenopterists are unable to identify most chalcidoids to species or genus level. Nevertheless, with the help of the specialists that contribute identifications on iNaturalist and [https://www.facebook.com/groups/HymenopteristsForum88/ Hymenopterists Forum], it has been possible to put together this page, illustrated with photographs of live wasps from 23 of the 35-odd chalcidoid families known in Africa; it is supplemented with photos of museum specimens from another four families.&lt;br&gt;
&lt;br&gt;
Many more images (mostly of museum specimens, including type specimens) can be found on [https://www.waspweb.org/Chalcidoidea/index.htm WaspWeb]. This includes photos of wasps from the following families not shown here: [https://www.waspweb.org/Chalcidoidea/Azotidae/index.htm Azotidae], [https://www.waspweb.org/Chalcidoidea/Cleonymidae/index.htm Cleonymidae], [https://www.waspweb.org/Chalcidoidea/Herbertiidae/index.htm Herbertiidae], [https://www.waspweb.org/Chalcidoidea/Heydeniidae/index.htm Heydeniidae], [https://www.waspweb.org/Chalcidoidea/Neanastatidae/index.htm Neanastatidae], and [https://www.waspweb.org/Chalcidoidea/Tanaostigmatidae/index.htm Tanaostigmatidae].&lt;br&gt;
&lt;br&gt;


===[[w:Agaonidae|Agaonidae]]===
&lt;gallery mode=packed heights=200&gt;
Agaonidae_2019_10_02_10_18_49_5304.jpg |''Ceratosolen galili'' female
Agaonidae 2019 10 10 10 17 37 3249b.jpg |''Elisabethiella comptoni'' female
Agaonidae 2019 10 10 10 46 58 3485.jpg |''Elisabethiella comptoni'' male
&lt;/gallery&gt;

===[[w:Aphelinidae|Aphelinidae]]===
&lt;gallery mode=packed heights=200&gt;
Aphelinidae 2022 05 19 3300.jpg|''[[w:Eretmocerus|Eretmocerus]]'' sp.
&lt;/gallery&gt;

===[[w:Calesidae|Calesidae]]===
&lt;gallery mode=packed heights=200&gt;
Cales 2023 07 01 iN 171704435.jpg|''Cales'' sp.; female
Cales 2023 07 01 iN 171706452.jpg|''Cales'' sp.; male
&lt;/gallery&gt;

===[[w:Ceidae|Ceidae]]===
&lt;gallery mode=packed heights=200&gt;
Bohpa 01 Mitroiu 2016.jpg|''Bohpa maculata'', female
Bopha 02 Mitroiu 2016.jpg|''Bohpa maculata'', female
&lt;/gallery&gt;

===[[w:Cerocephalidae|Cerocephalidae]]===
&lt;gallery mode=packed heights=200&gt;
Cerocephalidae 2023 06 05 iNat166277117 01.jpg
Cerocephalidae 2023 06 05 iNat166277117 02.jpg
&lt;/gallery&gt;

===[[:w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp.
Dirhinus 2019 07 28 7032.jpg|''Dirhinus'' sp.
Epitranus 2019 06 05 1831 Ced.jpg |''Epitranus'' sp.
Hockeria sp inat140473878 Jonathan Whitaker Oct 29, 2022.jpg

&lt;/gallery&gt;

===[[:w:Chrysolampidae|Chrysolampidae]]===
&lt;gallery mode=packed heights=200&gt;
Aperilampus sp.00.jpg|''Aperilampus'' sp., male
Aperilampus 2019 05 09 7697.jpg|''Aperilampus'' sp., female
&lt;/gallery&gt;

===[[:w:Diparidae|Diparidae]]===
&lt;gallery mode=packed heights=200&gt;
Conophorisca littoriticus inaturalist 356775210 2.jpg|''Conophorisca littoriticus''
Diparidae 2022 11 02 8329.jpg|Probably ''Lelaps'' sp. or ''Dipara'' sp. (male)
&lt;/gallery&gt;

===[[:w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Aenasius 2019 08 25 9691.jpg |''Aenasius'' sp.
Homalotylus 2019 08 16 8521.jpg|''Homalotylus'' sp.
&lt;/gallery&gt;

===[[:w:Eunotidae|Eunotidae]]===
&lt;gallery mode=packed heights=200&gt;
Scutellista iN 249602134 2024 10 02 11 57 50 4478.jpg|''Scutellista'' sp. at a ''Croton megalabotrys'' nectary
Scutellista iN 249602134.jpg|''Scutellista'' sp. on a ''Croton megalabotrys'' petiole
&lt;/gallery&gt;

===[[:w:Epichrysomallidae|Epichrysomallidae]]===
&lt;gallery mode=packed heights=200&gt;
Lachaisea brevimucro 2022 06 26 11 06 44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;

===[[:w:Eucharitidae|Eucharitidae]]===
&lt;gallery mode=packed heights=200&gt;
Hydrorhoa stevensoni iNat139715984 2022 10 12 5521.jpg|''Hydrorhoa stevensoni'', male
Stilbula 2020 01 02 9092.jpg |''Stilbula'' sp., female
Mateucharis glabra 2022 05 19 3435.jpg|''Mateucharis glabra'',  male
&lt;/gallery&gt;

===[[:w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Eulophinae 2019 06 10 16 16 23 1167.jpg |Subfamily Eulophinae (Unidentified)
Eulophidae 2025-09-21 inaturalist 315779038 01.jpg|''Euplectromorpha variegata''
Quadrastichus gallicola 172186279.jpg|''Quadrastichus gallicola''
Pediobius 2020 01 18 5090 PPot.jpg |''Pediobius'' sp. 
Tetrastichinae 2019 06 04 1502.jpg |Subfamily Tetrastichinae (Unidentified)
Neotrichoporoides 2019 08 20 8905.jpg |''Neotrichoporoides'' sp.
Neotrichoporoides 2019 06 28 4331.jpg |''Neotrichoporoides'' sp.
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp.
&lt;/gallery&gt;

===[[:w:Eupelmidae|Eupelmidae]]===
&lt;gallery mode=packed heights=200&gt;
Pentacladia 2019 08 04 7557.jpg |''Pentacladia'' sp.
Eupelmus 2019 08 16 8466.jpg |''Eupelmus'' sp.
Eupelmidae 2020 01 01 15 26 12 7925 G.jpg |''Eupelmidae'' sp.
&lt;/gallery&gt;

===[[:w:Eurytomidae|Eurytomidae]]===
&lt;gallery mode=packed heights=200&gt;
Eurytomidae00.jpg|Subfamily Eurytominae (Unidentified)
Sycophila 2019 07 06 0705.jpg |''Sycophila'' sp.
Sycophila 2019 08 24.jpg |''Sycophila'' sp.
Eurytominae 2019 10 08 17 12 49 0944Shi.jpg |Subfamily Eurytominae (Unidentified)
&lt;/gallery&gt;

===[[:w:Leucospidae|Leucospidae]]===
&lt;gallery mode=packed heights=200&gt;
Micrapion 2019 07 28 6711.jpg |''Micrapion'' sp.
Leucospis 2019 09 03 1195.jpg |''Leucospis'' sp.
&lt;/gallery&gt;

===[[:w:Lyciscidae|Lyciscidae]]===
&lt;gallery mode=packed heights=200&gt;
Solenura 2019 10 08 0781Shi.jpg|''Solenura nigra''
Solenura 2019 10 08 0786Shi.jpg|''Solenura nigra''
&lt;/gallery&gt;

===[[:w:Megastigmidae|Megastigmidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Megastigmus.jpg|''Megastigmus transvaalensis''
&lt;/gallery&gt;

===[[:w:Metapelmatidae|Metapelmatidae]]===
&lt;gallery mode=packed heights=200&gt;
Metapelma 2019 12 24 3471 C.jpg |''Metapelma'' sp.
&lt;/gallery&gt;

===[[:w:Mymaridae|Mymaridae]]===
&lt;gallery mode=packed heights=200&gt;
Mymaridae 2021 12 12 10 44 20.jpg|''Polynema'' sp.
Mymaridae 2022 05 15 12 32 49.jpg|''Polynema'' sp. or ''Acmopolynema'' sp.
Polynema sagittaria (10.3897-zookeys.783.26872) Figure 1A.jpg |''Polynema sagittaria''
&lt;/gallery&gt;

===[[:w:Ormyridae|Ormyridae]]===
&lt;gallery mode=packed heights=200&gt;
Ormyrus males 2024 09 28 iN 245187710 04.jpg|Male ''Ormyrus'' wasps on a stem gall
Ormyrus iNat90764688.jpg|Female ''Ormyrus'' wasp ovipositing
&lt;/gallery&gt;

===[[:w:Perilampidae|Perilampidae]]===
&lt;gallery mode=packed heights=200&gt;
Perilampus 2019 07 09 15 00 54 1165.jpg|''Perilampus'' sp.
&lt;/gallery&gt;

===[[:w:Pirenidae|Pirenidae]]===
&lt;gallery mode=packed heights=200&gt;
File:Pirenidae 160886456 04.jpg|''Spathopus'' sp.
&lt;/gallery&gt;

===[[:w:Pteromalidae|Pteromalidae ]]===
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg |''Halticoptera'' sp.
Philocaenus_rotundus_2019_10_10_3433.jpg|''Seres rotundus''
Otitesella 2019 08 25 32467865 9382.jpg|''Otitesella'' sp.
&lt;/gallery&gt;

===[[:w:Signiphoridae|Signiphoridae]]===
&lt;gallery mode=packed heights=200&gt;
Signiphora flavella Woolley &amp; Dal Molin 2017.jpg|''Signiphora flavella''
&lt;/gallery&gt;

===[[:w:Spalangiidae|Spalangiidae]]===
&lt;gallery mode=packed heights=200&gt;
Spalangia 2019 11 22 1217 Cedara.jpg |''Spalangia'' sp.
&lt;/gallery&gt;

===[[:w:Systasidae|Systasidae]]===
&lt;gallery mode=packed heights=200&gt;
Systasis 29313948-78.jpg|''Systasis'' sp.
Systasis 38824978-07.jpg|''Systasis'' sp.
&lt;/gallery&gt;

===[[:w:Tetracampidae|Tetracampidae]]===
&lt;gallery mode=packed heights=200&gt;
Afrocampe prinslooi Gumovsky 2018.jpg|''Afrocampe prinslooi''
&lt;/gallery&gt;

===[[:w:Torymidae|Torymidae ]]===
&lt;gallery mode=packed heights=200&gt;
Torymidae_2019_12_25_3502.jpg |Family Torymidae (unidentified)
Podagrion 2019 07 03 5701.jpg |''Podagrion'' sp.
Ecdamua 2019 12 30 7386 Cedara.jpg |''Ecdamua'' sp.
Pseudotorymus_2019_07_28_6571.jpg |''Pseudotorymus'' sp.
&lt;/gallery&gt;

===[[:w:Trichogrammatidae|Trichogrammatidae]]===
&lt;gallery mode=packed heights=200&gt;
Poropoea africana Laudonia 2017.jpg|''Poropoea africana''
&lt;/gallery&gt;

==References==
{{reflist}}

[[Category:African Arthropods]]</text>
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      <comment>project box(es)</comment>
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==Introduction==

Welcome to BSMQ Wikiversity resource for grade 12 learners. This resource consist of questions from almost all past years Business studies question papers in a Wikiversity quizzes form. Although they are mixed questions, this resource is design to resemble a single BS question papers as prepared by the department of education. If this resource is not enough, please visit the department of education website to download real question papers.

==Question 1 : Macro-environment==

Section A : Multiple choice


Tap next to the ✍️ imoji to answer and click on submit text bellow when you are done.

&lt;div style=&quot; border : 1px solid black&quot;&gt;
&lt;quiz display=simple points=&quot;2/1!&quot;&gt;

{
|type=&quot;{}&quot; coef=&quot;2&quot;}

&lt;b&gt;
1.Stipulates laws for workers conditions in a workplace.
&lt;/b&gt;

A. Basic Conditions of Employment Act.

B. Employment Equity Act.

C. Compensation of workplace Injury, Disease and Death Act.

D. Credit Act.


 ✍️ { A dog I big for a house (i) _23 }

&lt;b&gt;
2. It deals with hiring of employees.
&lt;/b&gt;


A. Marketing function

B. Production function

C. Human resource function

D. Financial function

 ✍️ { C (i) _1}


&lt;/quiz&gt;
&lt;/div&gt;
[[Category:Business]]</text>
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      <contributor>
        <username>CommonsDelinker</username>
        <id>9184</id>
      </contributor>
      <comment>Replacing President_Ronald_Reagan_working_at_his_desk_in_the_Oval_Office.jpg with [[File:President_Ronald_Reagan_Working_at_His_Desk_in_The_Oval_Office_-_DPLA_-_0bed851799b452dabc3d1dc32b77eb6b.jpg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because</comment>
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      <text bytes="538" sha1="1ehbop8zlhnwx50b6ch8zf8kc5fibr4" xml:space="preserve">{{Center|{{Quote|Freedom is never more than one generation away from extinction.|author=[[Ronald Reagan]]}}}}


{{Center|[[File:President Ronald Reagan Working at His Desk in The Oval Office - DPLA - 0bed851799b452dabc3d1dc32b77eb6b.jpg]]}}

{{Center|{{Caption|President Ronald Reagan working in the Oval Office, 15 July 1988}}}}


{{Center|Figuring out this page for now. I'll slowly add to this once I figure out this wiki.}}


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    <title>600-cell</title>
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      <comment>/* Hexagons and &lt;s&gt;hexagrams&lt;/s&gt; */ dodecagram should replace hexagram everywhere, since we discovered that the Clifford polygon of the hexagonal rotation is a {12/5} dodecagram, not a {6/2} hexagram as we previously stated -- this update is in progress in [[24-cell]] but has not been begun in this article yet</comment>
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      <text bytes="230834" sha1="m9w0eyk9qb6mnv96d0mgskoj3kranwp" xml:space="preserve">{{Short description|Four-dimensional analog of the icosahedron}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope |
  Name=600-cell|
  Image_File=Schlegel_wireframe_600-cell_vertex-centered.png|
  Image_Caption=[[W:Schlegel diagram|Schlegel diagram]], vertex-centered&lt;br&gt;(vertices and edges)|
  Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]|
  Last=[[W:Rectified 600-cell|34]]|
  Index=35|
  Next=[[W:Truncated 120-cell|36]]|
  Schläfli={3,3,5}|
  CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}|
  Cell_List=600 ([[W:Tetrahedron|{3,3}]]) [[Image:Tetrahedron.png|20px]]|
  Face_List=1200 [[W:triangle|{3}]]|
  Edge_Count=720|
  Vertex_Count= 120|
  Petrie_Polygon=[[W:Triacontagon#Petrie polygons|30-gon]]|
  Coxeter_Group=H&lt;sub&gt;4&lt;/sub&gt;, [3,3,5], order 14400|
  Vertex_Figure=[[Image:600-cell verf.svg|80px]]&lt;br&gt;[[W:icosahedron|icosahedron]]|
  Dual=[[120-cell|120-cell]]|
  Property_List=[[W:Convex polytope|convex]], [[W:isogonal figure|isogonal]], [[W:isotoxal figure|isotoxal]], [[W:isohedral figure|isohedral]]
}}
[[File:600-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[geometry]], the '''600-cell''' is the [[W:convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,5}.
It is also known as the '''C&lt;sub&gt;600&lt;/sub&gt;''', '''hexacosichoron'''&lt;ref&gt;[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249&lt;/ref&gt; and '''hexacosihedroid'''.&lt;ref&gt;Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68&lt;/ref&gt;
It is also called a '''tetraplex''' (abbreviated from &quot;tetrahedral complex&quot;) and a '''[[W:polytetrahedron|polytetrahedron]]''', being bounded by tetrahedral [[W:Cell (geometry)|cells]].

The 600-cell's boundary is composed of 600 [[W:Tetrahedron|tetrahedral]] [[W:Cell (mathematics)|cells]] with 20 meeting at each vertex.{{Efn|name=vertex icosahedral pyramid}}
Together they form 1200 triangular faces, 720 edges, and 120 vertices.
It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:icosahedron|icosahedron]], since it has five [[W:Tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[W:triangle|triangle]]s meeting at every vertex.{{Efn|name=math of dimensional analogy}}
Its [[W:dual polytope|dual polytope]] is the [[120-cell|120-cell]].

== Geometry ==

The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius).{{Efn|name=4-polytopes ordered by size and complexity|group=}}
It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the [[24-cell|24-cell]],{{Sfn|Coxeter|1973|loc=§8.51|p=153|ps=; &quot;In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s.&quot;}} as the 24-cell can be [[24-cell#8-cell|deconstructed]] into three overlapping instances of its predecessor the [[W:Tesseract|tesseract (8-cell)]], and the 8-cell can be [[24-cell#Relationships among interior polytopes|deconstructed]] into two instances of its predecessor the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}}

The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.
The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius,{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii), &quot;600-cell&quot; column &lt;sub&gt;0&lt;/sub&gt;''R/l'' {{=}} 2𝝓/2}} which is the [[W:golden ratio|golden ratio]].

{{Regular convex 4-polytopes|wiki=W:}}

=== Coordinates ===

==== Unit radius Cartesian coordinates ====
The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; (where &lt;math&gt;\phi = \tfrac12\bigl(1 + \sqrt5~\!\bigr)&lt;/math&gt; is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows:

8 vertices obtained from

:(0, 0, 0, ±1)

by permuting coordinates, and 16 vertices of the form:

:(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})

The remaining 96 vertices are obtained by taking [[W:even permutation|even permutation]]s of

:(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ&lt;sup&gt;−1&lt;/sup&gt;|2}}, 0)

Note that the first 8 are the vertices of a [[16-cell|16-cell]], the second 16 are the vertices of a [[W:tesseract|tesseract]], and those 24 vertices together are the vertices of a [[24-cell|24-cell]].
The remaining 96 vertices are the vertices of a [[W:snub 24-cell|snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}

When interpreted as [[#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[W:icosian|icosian]]s.

In the 24-cell, there are [[24-cell#Great squares|squares]], [[24-cell#Great hexagons|hexagons]] and [[24-cell#Great triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}}
In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint.
In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}}
In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}
Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]].

The 60 axes and 75 16-cells of the 600-cell constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60&lt;sub&gt;5&lt;/sub&gt;75&lt;sub&gt;4&lt;/sub&gt; to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's &quot;points&quot; and &quot;lines&quot; are axes (&quot;rays&quot;) and 16-cells (&quot;bases&quot;), respectively.}}
Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs.

==== Hopf spherical coordinates ====
In the 600-cell there are also great circle [[W:pentagon|pentagon]]s and [[W:decagon|decagon]]s (in central planes through ten vertices).{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020}}

Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell.
Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons.
Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|p=438}}
Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}}
Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon.
The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}}

By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} squares which do not share any vertices, or as 100 ''dual pairs'' of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex.
This latter pentagonal symmetry of the 600-cell is captured by the set of [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉&lt;sub&gt;''i''&lt;/sub&gt;, 𝜂, 𝜉&lt;sub&gt;''j''&lt;/sub&gt;){{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles:

: (𝜉&lt;sub&gt;''i''&lt;/sub&gt;, 𝜂, 𝜉&lt;sub&gt;''j''&lt;/sub&gt;)

that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles.
A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉&lt;sub&gt;''i''&lt;/sub&gt; and 𝜉&lt;sub&gt;''j''&lt;/sub&gt; are angles of rotation in the two [[W:completely orthogonal|completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]].
The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉&lt;sub&gt;''i''&lt;/sub&gt;, 0, 𝜉&lt;sub&gt;''j''&lt;/sub&gt;) coordinates describe the great circles which intersect at the north and south pole (&quot;lines of longitude&quot;).
The (𝜉&lt;sub&gt;''i''&lt;/sub&gt;, {{sfrac|𝜋|2}}, 𝜉&lt;sub&gt;''j''&lt;/sub&gt;) coordinates describe the great circles orthogonal to longitude (&quot;equators&quot;); there is more than one &quot;equator&quot; great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles.
The other Hopf coordinates (𝜉&lt;sub&gt;''i''&lt;/sub&gt;, 0 &lt; 𝜂 &lt; {{sfrac|𝜋|2}}, 𝜉&lt;sub&gt;''j''&lt;/sub&gt;) describe the great circles (''not'' &quot;lines of latitude&quot;) which cross an equator but do not pass through the north or south pole.}}
Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉&lt;sub&gt;''i''&lt;/sub&gt;, 𝜂, 𝜉&lt;sub&gt;''j''&lt;/sub&gt;) to unit-radius Cartesian coordinates (w, x, y, z) is:&lt;br&gt;
: w {{=}} cos 𝜉&lt;sub&gt;''i''&lt;/sub&gt; sin 𝜂
: x {{=}} cos 𝜉&lt;sub&gt;''j''&lt;/sub&gt; cos 𝜂
: y {{=}} sin 𝜉&lt;sub&gt;''j''&lt;/sub&gt; cos 𝜂
: z {{=}} sin 𝜉&lt;sub&gt;''i''&lt;/sub&gt; sin 𝜂
The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional &quot;north pole&quot; of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as:

: ({&lt;10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {&lt;10}{{sfrac|𝜋|5}})

where {&lt;10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5).
The 𝜉&lt;sub&gt;''i''&lt;/sub&gt; and 𝜉&lt;sub&gt;''j''&lt;/sub&gt; coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.{{Efn|There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell.
These are actually the Hopf coordinates of the vertices of the [[120-cell#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}}

=== Structure ===

==== Polyhedral sections ====
The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[W:hypersphere|hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋.
Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[W:icosahedron|icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[W:dodecahedron|dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[W:icosidodecahedron|icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex}}{{Sfn|van Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout &amp; Parker|1998}}
These can be seen in the H3 [[W:Coxeter plane|Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}

:[[File:600-cell-polyhedral levels.png|640px]]

These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid).
Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane).
In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center.
But its own center is in the interior of the 600-cell, not on its surface.
V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell.
Thus V is the apex of a [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron.

{| class=wikitable
!colspan=2|Concentric Hulls
|-
|align=center|[[Image:Hulls of H4only-orthonormal.png|360px]]
|The 600-cell is projected to 3D using an orthonormal basis.
The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:&lt;br&gt;
&lt;br&gt;
1) two points at the origin&lt;br&gt;
2) two icosahedra&lt;br&gt;
3) two dodecahedra&lt;br&gt;
4) two larger icosahedra&lt;br&gt;
5) and a single icosidodecahedron&lt;br&gt;
&lt;br&gt;
for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices.
|-
|}

==== Golden chords ====
[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]]
The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]].
The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc.
The golden ratio governs{{Efn|name=golden chords|group=}} the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.{{Efn|name=radially golden}}|alt=|400x400px]]
{{see also|W:24-cell#Hypercubic chords|label 1=24-cell § Hypercubic chords}}
The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[W:Chord (geometry)|chord]] lengths from each other.
These edges and chords of the 600-cell are simply the edges and chords of its [[#Geodesics|five great circle polygons]].{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23|loc=Figure 3}}
In ascending order of length, they are:

 &lt;math&gt;\sqrt{0.382\sim} = \sqrt{2 - \phi} = \phi^{-1} \approx 0.618&lt;/math&gt;
 &lt;math&gt;\sqrt{1}&lt;/math&gt;
 &lt;math&gt;\sqrt{1.382\sim} = \sqrt{3 - \phi} \approx 1.176&lt;/math&gt;
 &lt;math&gt;\sqrt{2}&lt;/math&gt;
 &lt;math&gt;\sqrt{2.618\sim} = \sqrt{1 + \phi} = \phi \approx 1.618&lt;/math&gt;
 &lt;math&gt;\sqrt{3}&lt;/math&gt;
 &lt;math&gt;\sqrt{3.618\sim} = \sqrt{2 + \phi} \approx 1.902&lt;/math&gt;
 &lt;math&gt;\sqrt{4}&lt;/math&gt;

In the diagram, chord lengths are given as square roots, with a decimal fractional part if necessary, where:
 &lt;math&gt;\Phi = \phi^{-1} \approx 0.618&lt;/math&gt; 
is the inverse golden ratio, and:
 &lt;math&gt;\Delta = 1 - \Phi = \Phi^2 \approx 0.382&lt;/math&gt;
is its square. For example, the 600-cell edge length is:
 &lt;math&gt;\Phi = \sqrt{0.\Delta} = \sqrt{0.382\sim} \approx 0.618&lt;/math&gt;
The four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell (&lt;math&gt;\sqrt{1}&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;\sqrt{3}&lt;/math&gt;, &lt;math&gt;\sqrt{4}&lt;/math&gt;){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons.
The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of &lt;math&gt;\sqrt{5}&lt;/math&gt;.{{Efn|The squares of two of these chord lengths, &lt;math&gt;3 - \phi {{=}} \phi^{-1}\sqrt{5}&lt;/math&gt; and &lt;math&gt;2 + \phi {{=}} \phi\sqrt{5}&lt;/math&gt;, are [[W:Algebraic conjugate|algebraic conjugate]]s whose product is &lt;math&gt;5&lt;/math&gt;.}} The golden chords of the 600-cell exemplify that the [[W:golden ratio|golden ratio]] &lt;math&gt;\phi {{=}} \tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt; is a circle ratio related to fifths of &lt;math&gt;\pi&lt;/math&gt;. For instance:&lt;br&gt;
:&lt;math&gt;\tfrac{\pi}{5} {{=}} \arccos(\tfrac{\phi}{2})&lt;/math&gt;
is the arc of one 600-cell edge, the &lt;math&gt;\phi^{-1} = \Phi \approx 0.618&lt;/math&gt; chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;math&gt;\phi&lt;/math&gt; as a function of &lt;math&gt;\pi&lt;/math&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;math&gt;\phi {{=}} 1 - 2 \cos(\tfrac{3\pi}{5})&lt;/math&gt;
&lt;math&gt;\tfrac{3\pi}{5}&lt;/math&gt; is the arc length of the &lt;math&gt;\phi \approx 1.618&lt;/math&gt; chord.&lt;ref&gt;{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=W:John Carlos Baez|access-date=10 October 2022}}&lt;/ref&gt;|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of &lt;math&gt;\sqrt{5}&lt;/math&gt;.{{Efn|The 600-cell edges are decagon edges of length &lt;math&gt;\phi^{-1} {{=}} \Phi {{=}} \sqrt{0.\Delta} \approx 0.618&lt;/math&gt;, the ''smaller'' golden section of &lt;math&gt;\sqrt{5}&lt;/math&gt;; the edges are in the inverse [[W:golden ratio|golden ratio]] &lt;math&gt;\tfrac{1}{\phi} {{=}} \phi^{-1}&lt;/math&gt; to the &lt;math&gt;\sqrt{1}&lt;/math&gt; hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length &lt;math&gt;\sqrt{3 - \phi} {{=}} \sqrt{1.\Delta}&lt;/math&gt; is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length &lt;math&gt;\phi {{=}} \sqrt{2.\Phi}&lt;/math&gt;, the ''larger'' golden section of &lt;math&gt;\sqrt{5}&lt;/math&gt;; it is in the golden ratio{{Efn|name=golden chords|group=}} to the &lt;math&gt;\sqrt{1}&lt;/math&gt; chord (and the radius).{{Efn|Notice in the diagram how the &lt;math&gt;\phi&lt;/math&gt; chord (the ''larger'' golden section) sums with the adjacent &lt;math&gt;\Phi&lt;/math&gt; edge (the ''smaller'' golden section) to &lt;math&gt;\sqrt{5}&lt;/math&gt;, as if together they were a &lt;math&gt;\sqrt{5}&lt;/math&gt; chord bent to fit inside the &lt;math&gt;\sqrt{4}&lt;/math&gt; diameter.}}
The last fractional-root chord is the pentagon diagonal of length &lt;math&gt;=\sqrt{2 + \phi} {{=}} \sqrt{3.\Phi}&lt;/math&gt;.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;math&gt;\sqrt{2 + \phi} / \sqrt{3 - \phi} {{=}} \phi&lt;/math&gt;.|name=fractional root chords|group=}}

==== Boundary envelopes ====
[[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].
The 3D surface made of 600 tetrahedra is visible.]]
The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell.
The other 96 vertices constitute a [[W:snub 24-cell|snub 24-cell]].
Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}}
The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells.
One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;.
It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the [[24-cell#Hypercubic chords|&lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords]].

[[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell|24-cell]] performing a [[24-cell#Simple rotations|simple rotation]].
The 3D surface made of 24 octahedra is visible.
It is also present in the 600-cell, but as an invisible interior boundary envelope.]]
Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[W:Tesseract#Radial equilateral symmetry|radially equilateral]].
Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[W:equilateral triangle|equilateral triangle]]s which all meet at the center.{{Efn|The long radius (center to vertex) of the [[24-cell#geometry|24-cell]] is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[W:Golden triangle (mathematics)|golden triangle]]s which all meet at the center.{{Efn|A [[W:Golden triangle (mathematics)|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ratio|golden ratio]] to the distinct side ''b'':
:&lt;math&gt;\tfrac{a}{b} {{=}} \phi {{=}} \tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;
It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center, and in the regular [[W:Pentagon|pentagon]] by connecting any two adjacent vertices to the vertex opposite them.&lt;br&gt;
The vertex angle is:
:&lt;math&gt;\theta = \arccos(\tfrac{\phi}{2}) {{=}} \tfrac{\pi}{5} {{=}} 36^\circ&lt;/math&gt;
so the base angles are each &lt;math&gt;\tfrac{2\pi}{5} {{=}} 72^\circ&lt;/math&gt;.
The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=Golden triangle}} All the [[W:regular polytope|regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[W:Schläfli orthoscheme|orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[W:right triangle|right triangle]]s which meet at the center.{{Efn|The [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is the generalization of the [[W:right triangle|right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[W:golden ratio|golden ratio]] to its edge length; thus its radius is &lt;math&gt;\phi&lt;/math&gt; if its edge length is 1, and its edge length is &lt;math&gt;\phi^{-1}&lt;/math&gt; if its radius is 1.|name=radially golden}}
Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[W:icosidodecahedron|icosidodecahedron]], and the two-dimensional [[W:Decagon#The golden ratio in decagon|decagon]].
(The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.)
'''Radially golden''' polytopes are those which can be constructed, with their radii, from [[W:Golden triangle (mathematics)|golden triangles]].{{Efn|name=Golden triangle}}

The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes).
The shape of those interstices must be an [[W:Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}}
Therefore the successor may be constructed by placing [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor.
Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[W:cubic pyramid|cubic pyramid]]s.
But if we place 24 canonical [[W:octahedral pyramid|octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell.
Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}}

==== Geodesics ====
The vertex chords of the 600-cell are arranged in [[W:geodesic|geodesic]] [[W:great circle|great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=§4 The planes of the 600-cell|pp=437-439}}

[[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[W:stereographic projection|stereographic projection]] of the 600-cell's 72 central decagons onto their great circles.
Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]]
The &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; edges form 72 flat regular central [[W:decagon|decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}}
Just as the [[W:icosidodecahedron|icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10).
The 720 &lt;math&gt;\phi^{-1}&lt;/math&gt; edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, &lt;math&gt;\sqrt{2 + \phi}&lt;/math&gt; apart.
As in the decagon and the icosidodecahedron, the edges occur in [[W:Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope.
The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[W:Hopf fibration|Hopf fibration]]s, each filling the whole 600-cell. Each [[#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}

The &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The &lt;math&gt;\sqrt{1}&lt;/math&gt; chords join vertices which are two &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart.
Each &lt;math&gt;\sqrt{1}&lt;/math&gt; chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[W:triangular bipyramid|triangular bipyramid]]), and passes through the center of the shared face.
As there are 1200 faces, there are 1200 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, in 600 parallel pairs, &lt;math&gt;\sqrt{3}&lt;/math&gt; apart.
The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.
The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs.
Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}}
The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}

The &lt;math&gt;\sqrt{3 - \phi} \approx 1.176&lt;/math&gt; chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}}
The &lt;math&gt;\sqrt{3 - \phi}&lt;/math&gt; chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon.
They join vertices which are two &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart on a geodesic great circle.
The 720 &lt;math&gt;\sqrt{3 - \phi}&lt;/math&gt; chords occur in 360 parallel pairs, &lt;math&gt;\phi&lt;/math&gt; apart.

The &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex).
The &lt;math&gt;\sqrt{2}&lt;/math&gt; chords join vertices which are three &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart (and two &lt;math&gt;\sqrt{1}&lt;/math&gt; chords apart).
There are 600 &lt;math&gt;\sqrt{2}&lt;/math&gt; chords, in 300 parallel pairs, &lt;math&gt;\sqrt{2}&lt;/math&gt; apart.
The 450 great squares (225 [[W:Completely orthogonal|completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}

The &lt;math&gt;\phi \approx 1.618&lt;/math&gt; chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex.
The third edge (base) of each isosceles triangle is of length &lt;math&gt;\sqrt{2 + \phi} \approx 1.902&lt;/math&gt;.
The &lt;math&gt;\phi&lt;/math&gt; chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart on a geodesic great circle.
There are 720 distinct &lt;math&gt;\phi&lt;/math&gt; chords, in 360 parallel pairs, &lt;math&gt;\sqrt{3 - \phi}&lt;/math&gt; apart.

The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells).
Each set of 32 triangles consists of the 96 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords join vertices which are four &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart (and two &lt;math&gt;\sqrt{1}&lt;/math&gt; chords apart on a geodesic great circle).
Each &lt;math&gt;\sqrt{3}&lt;/math&gt; chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 &lt;math&gt;\sqrt{1}&lt;/math&gt; cubic cells.
The 1200 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each &lt;math&gt;\sqrt{3}&lt;/math&gt; chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}}
There are 1200 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, in 600 parallel pairs, &lt;math&gt;\sqrt{1}&lt;/math&gt; apart.

The &lt;math&gt;\sqrt{2 + \phi} \approx 1.902&lt;/math&gt; chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex.
The third edge (base) of each isosceles triangle is an edge of the pentagon of length &lt;math&gt;\sqrt{3 - \phi} \approx 1.176&lt;/math&gt;, so these are [[W:Golden triangle (mathematics)|golden triangles]]. The &lt;math&gt;\sqrt{2 + \phi}&lt;/math&gt; chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart on a geodesic great circle.
There are 720 distinct &lt;math&gt;\sqrt{2 + \phi}&lt;/math&gt; chords, in 360 parallel pairs, &lt;math&gt;\phi^{-1}&lt;/math&gt; apart.

The &lt;math&gt;\sqrt{4}&lt;/math&gt; chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell.
The &lt;math&gt;\sqrt{4}&lt;/math&gt; chords join opposite vertices which are five &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart on a geodesic great circle.
There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells.

The sum of the squared lengths of all these distinct chords of the 600-cell is 14,400 = 120&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}} In this case, &lt;math&gt;(2 - \phi) \cdot 720 + 1 \cdot 1200 + {}\!&lt;/math&gt;&lt;math&gt;(3 - \phi) \cdot 720 + 2 \cdot 1800 + {}\!&lt;/math&gt;&lt;math&gt;(1 + \phi)\cdot 720 + 3\cdot 1200 + {}\!&lt;/math&gt;&lt;math&gt;(2 + \phi) \cdot 720 + 4 \cdot 60&lt;/math&gt; is 14,400.}} 
These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[W:completely orthogonal|completely orthogonal]] to a great 30-gon{{Efn|A ''[[W:triacontagon|triacontagon]]'' or 30-gon  is a thirty-sided polygon.
The triacontagon is the largest regular polygon whose interior angle is the sum of the [[W:Interior angle|interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[W:Equilateral triangle|equilateral triangle]] (60°) and the [[W:Regular pentagon|regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell.
The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly.
The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords.
It does not touch any edges or vertices, but it does hit faces.
It is the central axis of a spiral skew 30-gram, the [[W:Petrie polygon|Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}}
Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all.
There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}}

All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells).
These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:icosidodecahedron|icosidodecahedron]].
There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes.
(More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}}
Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space.
Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens &amp; Seidel|1973}}
Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together.
Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration.
Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles.
Vectors in 4-space may be combined by [[W:Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[W:William Rowan Hamilton|Hamilton]].{{Sfn|Mamone, Pileio &amp; Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0).
Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector.
Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors &lt;small&gt;&lt;math&gt;\left(w,x,y,z\right)_1&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\left(w,x,y,z\right)_2&lt;/math&gt;&lt;/small&gt; according to&lt;br&gt;
&lt;small&gt;&lt;math display=block&gt;\begin{pmatrix}
w_2\\
x_2\\
y_2\\
z_2
\end{pmatrix}
*
\begin{pmatrix}
w_1\\
x_1\\
y_1\\
z_1
\end{pmatrix}
=
\begin{pmatrix}
{w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\
{w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\
{w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\
{w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1}
\end{pmatrix}
&lt;/math&gt;&lt;/small&gt;}}
The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}}
Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}}
Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell.
Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}}
Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}}
Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}}
A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}}
Each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one &lt;math&gt;\sqrt{4}&lt;/math&gt; long diameter): a great [[W:digon|digon]] plane.{{Efn|In the 24-cell each great square plane is [[W:completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:digon|digon]] plane.|name=digon planes}}
Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane.
Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere.
Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere.
Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell.
In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}}

==== Fibrations of great circle polygons ====
Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}}
Each [[W:fiber bundle|fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[W:Hopf fibration|Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S&lt;sub&gt;3&lt;/sub&gt;|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}}
The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets.
The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different &quot;directions&quot;; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell &amp; Semple|1971|loc=§4. Isoclinic planes in Euclidean space E&lt;sub&gt;4&lt;/sub&gt;|pp=6-7}}

===== Decagons =====
[[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[W:Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[Schläfli double six|Schläfli double six]] configuration 30&lt;sub&gt;2&lt;/sub&gt;12&lt;sub&gt;5&lt;/sub&gt; characteristic of the H&lt;sub&gt;4&lt;/sub&gt; polytopes. The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]]
The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.

Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[W:icosahedron|icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}}
Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations.
The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}}

The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[W:Hopf fibration|Hopf fibration]]s make the 600-cell a [[W:Configuration (geometry)|geometric configuration]] of 30 &quot;points&quot; and 12 &quot;lines&quot; written as 30&lt;sub&gt;2&lt;/sub&gt;12&lt;sub&gt;5&lt;/sub&gt;.
It is called the [[Schläfli double six|Schläfli double six]] configuration after [[W:Ludwig Schläfli|Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[W:Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; &quot;The finite group [3&lt;sup&gt;2, 2, 1&lt;/sup&gt;] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)&quot;.}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}}

===== Hexagons =====
The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons.
Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon.
Each octahedral cell occupies only one cell ring in each of the 4 fibrations.
The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration.

The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}}
It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. 
The Hopf map of this fibration is the [[W:dodecahedron|dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}
Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations.
The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells.

===== Squares =====
The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares.
Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each.
Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations.
The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration.

The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells.
It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the &lt;math&gt;\sqrt{2}&lt;/math&gt; tetrahedral cells of the 75 inscribed 16-cells, ''not'' the &lt;math&gt;\phi^{-1}&lt;/math&gt; tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} 
The Hopf map of this fibration is the [[W:icosidodecahedron|icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}
Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations.

===== Clifford parallel cell rings =====
The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, studied cell rings in the general case of their geometry and [[W:group theory|group theory]], identifying each cell ring as a [[W:polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]].{{Efn|name=orthoscheme ring}}
He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:Chiral|chiral]] forms.
Specifically, he found that the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}}
Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S&lt;sub&gt;3&lt;/sub&gt;|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces.
Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}}
The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells.
They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}}
The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell).
The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell.
The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells.
In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}}
This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy &quot;always works&quot;, or if it is perhaps &quot;just guesswork&quot; that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[W:Hopf fibration|Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}}
The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings.
Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration).
In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}}
The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration.
All the fibrations have this two level structure with ''subfibrations''.

The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon.
Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.

The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square.
Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}}

The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope.
Each fibration is the entire 4-polytope.}}
The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[W:Hopf fibration|Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[W:2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}}
On this map each great circle fiber appears as a single point.
The base of a great decagon fibration of the 600-cell is the [[W:icosahedron|icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}
To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration.
But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[W:Vertex figure|vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell.
Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[W:Braid|braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell, ''generating'' the 600-cell.
Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration's characteristic isoclinic rotation generates the 600-cell, since the Hopf fibration is an expression of an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Efn|Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space, as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, &quot;frustrated&quot; by their torsion.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]]....{{Efn|name=Petrie polygon of a honeycomb}} The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;|name=Sadoc frustration}}|name=Hopf fibration base}} of these fibrations on the 2-sphere).
Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 {{Background color|red}} faces of the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration'').
The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}}

=== Constructions ===
The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}}
Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial.
The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize.

==== Gosset's construction ====
[[W:Thorold Gosset|Thorold Gosset]] discovered the [[W:Semiregular polytope|semiregular 4-polytopes]], including the [[W:Snub 24-cell|snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius.
Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form.
In the first, more complex step (described [[W:Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[W:Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}
In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}}

The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a &quot;flat&quot; icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}}
The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells.
The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.

Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps.
The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation.
Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[W:Midsphere|midsphere]] to construct its outer [[W:Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual.
That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell.

==== Cell clusters ====
Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]], or how they lie on the underlying surface envelope of the 24-cell's octahedral cells.
For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.{{Efn|name=tetrahedral cell adjacency}}

Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; &quot;The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences....
[Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains.
Coxeter made full use of it, and expected the reader to use it....
Visualization is one of the most powerful interiorization techniques.
It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module.
Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases.&quot;}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional honeycomb{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|p=19}} that we could actually &quot;walk around in&quot; and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}}
In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[W:Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.

===== Icosahedra =====
[[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[W:Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[W:Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[W:Regular icosahedron#Symmetries|snub octahedron]].
This term refers specifically to a [[W:Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}}
Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces.
The apex of the [[W:Icosahedral pyramid|icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible).
The four cells lie in different hyperplanes.|alt=|200x200px]]
The [[W:Vertex figure|vertex figure]] of the 600-cell is the [[W:Icosahedron|icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center.
Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there.
However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices.
Thus the vertex icosahedron is actually a canonical [[W:Icosahedral pyramid|icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|name=radially equilateral icosahedral pyramid}}|name=vertex icosahedral pyramid|group=}}
Twenty tetrahedral cells meet at each vertex, forming an [[W:Icosahedral pyramid|icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron.
It is remarkable that twenty regular tetrahedra fit inside a regular icosahedral pyramid in 4-space. In 3-space, twenty triangular pyramids fit inside a regular icosahedron around its center but they are ''not'' regular tetrahedra, because the regular icosahedron's radius is not the same as its edge length.{{Efn|In Euclidean 3-space, the icosahedron is not [[W:Cuboctahedron#Radial equilateral symmetry|radially equilateral like the cuboctahedron]]. The icosahedron's radii are shorter than its edge length. But in the [[W:3-sphere|spherical 3-space]] of the 600-cell's surface the center of a regular icosahedron is lifted orthogonally out of its 3-space hyperplane: remarkably, just far enough to make its radii the same length as its edges. As a figure in Euclidean 4-space, this radially equilateral spherical icosahedron is an [[W:Icosahedral pyramid|icosahedral pyramid]]. In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the icosahedral pyramid is not its center, just one of its vertices. But in curved 3-space the 12 edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that spherical space''. In Euclidean 4-space there are only two radially equilateral figures: 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[W:tesseract#Radial equilateral symmetry|radially equilateral tesseract]].|name=radially equilateral icosahedral pyramid}}
The 600-cell has a [[W:Dihedral angle|dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}}

An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra.
Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five).
Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells.
Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}}

The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell.
The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[W:Snub 24-cell|snub 24-cell]], which has exactly the same [[W:Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only &quot;flat&quot; 3-dimensional icosahedral cells, because the central apical vertex is missing.

The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces.
Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the &quot;[[W:Regular icosahedron#Symmetries|snub octahedron]]&quot; icosahedron all look the same, with two yellow and three blue faces.
Each pentagon has five distinct rotational orientations.
Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}}
Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell.
Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}

The icosahedra are face-bonded into geodesic &quot;straight lines&quot; by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids.
Their apexes are the vertices of a [[24-cell#Great hexagons|great circle hexagon]].
This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[W:Triangular bipyramid|triangular bipyramid]]) is a hexagon edge (a 24-cell edge).
There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a &lt;math&gt;\sqrt{1}&lt;/math&gt;octahedral cell, but in the larger &lt;math&gt;\sqrt{2}&lt;/math&gt; octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron.
The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]].
The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}}
The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[W:Russian dolls|Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}}
The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}}
The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[W:Jessen's icosahedron|Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 6 vertices of the octahedron;{{Sfn|Itoh &amp; Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; &quot;This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[W:Jitterbug transformation|Jitterbug]] by [[W:Buckminster Fuller|Buckminster Fuller]].&quot;}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}}
The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[W:Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca et. al.|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[W:3-sphere|S&lt;sup&gt;3&lt;/sup&gt;]] is similar to the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]] and the [[W:Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[W:Three-dimensional space|R&lt;sup&gt;3&lt;/sup&gt;]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[W:Buckminster Fuller|Buckminster Fuller]].&lt;ref&gt;{{cite journal
 | last = Verheyen | first = H. F.
 | doi = 10.1016/0898-1221(89)90160-0
 | issue = 1–3
 | journal = [[W:Computers and Mathematics with Applications|Computers and Mathematics with Applications]]
 | mr = 0994201
 | pages = 203–250
 | title = The complete set of Jitterbug transformers and the analysis of their motion
 | volume = 17
 | year = 1989| doi-access = free
 }}&lt;/ref&gt;}}

The tetrahedral cells are face-bonded into [[W:Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[W:Boerdijk-Coxeter helix|Boerdijk-Coxeter helix]].
This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]].
In the 600-cell we find them bent in the fourth dimension into geodesic rings.
Each ring has 30 cells and touches 30 vertices.
The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}}
5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge).
A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[W:Hopf fibration|Hopf fibration]].
There are 6 distinct such Hopf fibrations, covering the same space but running in different &quot;directions&quot;.|name=Boerdijk–Coxeter helix}}
The three helices are geodesic &quot;straight lines&quot; of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other.
Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex.
Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[W:Skew lines|skew edges]] of the tetrahedron.
Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron.
However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[W:Hopf fibration|Hopf fiber bundle]] of 12.
Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]].

The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same.
One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}}
Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}}

===== Octahedra =====
There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell.

Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells.
The central cell is the first section of the 600-cell beginning with a cell.
By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell.

First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[W:Triangular dipyramid|triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length &lt;math&gt;\sqrt{1}&lt;/math&gt;
Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs.
They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length &lt;math&gt;\sqrt{1}&lt;/math&gt;.
They form a tetrahedron of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;, which is the second section of the 600-cell beginning with a cell.{{Efn|The &lt;math&gt;\sqrt{1}&lt;/math&gt; tetrahedron has a volume of 9 &lt;math&gt;\phi^{-1}&lt;/math&gt; tetrahedral cells.
In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it.
The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the &lt;math&gt;\sqrt{1}&lt;/math&gt; tetrahedron.
The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}}
There are 600 of these &lt;math&gt;\sqrt{1}&lt;/math&gt; tetrahedral sections in the 600-cell.

With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster.
The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}}
As partially filled so far (by 17 tetrahedral cells), this &lt;math&gt;\sqrt{1}&lt;/math&gt; octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each &lt;math&gt;\sqrt{1}&lt;/math&gt; edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells).
In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces.
Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}}
Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A &lt;math&gt;\sqrt{1}&lt;/math&gt; octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell.
The same &lt;math&gt;\sqrt{1}&lt;/math&gt; octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point &lt;math&gt;\sqrt{1}&lt;/math&gt; octahedral section, a 4-point &lt;math&gt;\sqrt{1}&lt;/math&gt; tetrahedral section, and a 4-point &lt;math&gt;\phi^{-1}&lt;/math&gt; tetrahedral section.
In the curved three-dimensional space of the 600-cell's surface, the &lt;math&gt;\sqrt{1}&lt;/math&gt; octahedron surrounds the &lt;math&gt;\sqrt{1}&lt;/math&gt; tetrahedron which surrounds the &lt;math&gt;\phi^{-1}&lt;/math&gt; tetrahedron, as three concentric hulls.
This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[W:Octahedral pyramid|octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}}

Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical &lt;math&gt;\sqrt{1}&lt;/math&gt; [[W:Octahedral pyramid|octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter &lt;math&gt;\phi^{-1}&lt;/math&gt; edges, replacing the apex with four vertices.
The truncation has also created another four vertices (arranged as a &lt;math&gt;\sqrt{1}&lt;/math&gt; tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with &lt;math&gt;\phi^{-1}&lt;/math&gt; edges.
The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all.
The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two &lt;math&gt;\sqrt{1}&lt;/math&gt; edges (and just one of those routes ran through the single apex).
The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three &lt;math&gt;\phi^{-1}&lt;/math&gt; edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[W:Rectified 5-cell|rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length &lt;math&gt;\phi^{-1}&lt;/math&gt;.

===== Union of two tori =====
There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; &quot;Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope.&quot;}} and the [[#Decagons|decagonal fibrations]] of the 600-cell.
An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic &quot;straight lines&quot;.

[[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[W:Clifford torus|Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}}
Its opposite edges are identified, forming a [[W:Duocylinder|duocylinder]].]]
The [[120-cell|120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]].
Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex.
The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}

Start by assembling five tetrahedra around a common edge.
This structure looks somewhat like an angular &quot;flying saucer&quot;.
Stack ten of these, vertex to vertex, &quot;pancake&quot; style.
Fill in the annular ring between each pair of &quot;flying saucers&quot; with 10 tetrahedra to form an icosahedron.
You can view this as five vertex-stacked [[W:Icosahedral pyramids|icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet.
This 10-cell ring is shaped like a [[W:Pentagonal antiprism|pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center.
This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}}
Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}}
The surface is the same as that of ten stacked [[W:pentagonal antiprism|pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}}
Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices.
Stack another tetrahedron on each exposed face.
This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[W:Triangular bipyramid|triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge.
The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}}
The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons).
These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes).

Build a second identical torus of 250 cells that interlinks with the first.
This accounts for 500 cells.
These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges.
This latter set of 100 tetrahedra are on the exact boundary of the [[W:Duocylinder|duocylinder]] and form a [[W:Clifford torus|Clifford torus]].{{Efn|A [[W:Clifford torus|Clifford torus]] is the [[W:Hopf fibration|Hopf fiber bundle]] of a distinct [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[W:3-sphere|3-sphere]], involving all of its points. The [[W:SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[W:Cartesian product|Cartesian product]] of two [[W:Completely orthogonal|completely orthogonal]] [[W:Great circle|great circle]]s. It is a filled [[W:Doughnut|doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[W:4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}}
They can be &quot;unrolled&quot; into a square 10×10 array.
Incidentally this structure forms one tetrahedral layer in the [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]].
There are exactly 50 &quot;egg crate&quot; recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy &quot;egg crate&quot; square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} How can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[W:Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}}
In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[W:Triangular bipyramid|triangular bipyramid]] composed of two tetrahedra.

This decomposition of the 600-cell has [[W:Coxeter notation|symmetry]] [10,2&lt;sup&gt;+&lt;/sup&gt;,10], order 400, the same symmetry as the [[W:Grand antiprism|grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H&lt;sub&gt;2&lt;/sub&gt; × H&lt;sub&gt;2&lt;/sub&gt;}}
The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[W:Pentagonal antiprism|pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}}

The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells.

===== Boerdijk–Coxeter helix rings =====
The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[W:Hopf fibration|Hopf fibration]] which fills the entire 600-cell.{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}
Each ring of 30 face-bonded tetrahedra is a cylindrical [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension.

{| class=&quot;wikitable&quot; width=&quot;600&quot;
|[[File:600-cell tet ring.png|200px]]&lt;br&gt;A single 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}}
|[[File:600-cell Coxeter helix-ring.png|200px]]&lt;br&gt;A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}}
|[[File:Regular_star_polygon_30-11.svg|200px]]&lt;br&gt;The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[W:Skew polygon|skew]] {30/11} [[W:Star polygon|star polygon]] with a [[W:Winding number|winding number]] of 11 called a [[W:Triacontagon#Triacontagram|triacontagram&lt;sub&gt;11&lt;/sub&gt;]], a continuous tight corkscrew [[W:Helix|helix]] bent into a loop of 30 edges (the {{Background color|magenta|magenta}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[W:Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[W:Density (polytope)|characterized]] as the [[W:Petrie polygon|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}}
|-
|colspan=3|[[File:Coxeter_helix_edges.png|625px]]&lt;br&gt;The 30-vertex, 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''.
|}

The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three {{Background color|cyan|cyan}} Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}}
The 30 {{Background color|magenta|magenta}} edges joining these vertex pairs form a helical [[W:Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[W:Petrie polygon|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[W:Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[W:Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h&lt;sub&gt;1&lt;/sub&gt; h&lt;sub&gt;2&lt;/sub&gt;''.}}|name=Petrie polygon in 30-cell ring}}
The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[W:Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell|120-cell]], the 600-cell's [[W:Dual polytope|dual polytope]].{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the 600-cell and its dual the [[120-cell|120-cell]]. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}}
The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}}

The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices.
Each orange or yellow edge crosses between two {{Background color|cyan|cyan}} great decagons.
Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}}
Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[W:Geodesic|geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}}
The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between.
A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop.
The two connected loops forms a single [[W:Möbius loop|Möbius loop]], a skew {15/2} [[W:Pentadecagram|pentadecagram]].
The pentadecagram is not shown in these illustrations (but [[#Decagons and pentadecagrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations.
Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}}
Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}}
They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[W:link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}}

Five of these 30-cell [[W:Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H&lt;sub&gt;2&lt;/sub&gt; × H&lt;sub&gt;2&lt;/sub&gt;}}
Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[W:Chiral|chiral]] objects; they either spiral clockwise (right) or counterclockwise (left).
The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings.
Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[W:Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]].
Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration.
In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings.
Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[W:Hopf link|Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Efn|Sadoc describes the decomposition of the 600-cell into four tori.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} It is the same [[#Decagons|fibration of 12 great decagons and 20 30-cell rings]], seen as a [[#Clifford parallel cell rings|fibration of four completely disjoint 30-cell rings]]{{Efn|name=completely disjoint}} with spaces between them, which still encompasses all 12 decagons and all 120 vertices. If we look closely at the spaces between the four disjoint 30-cell rings, we ''can'' discern four 150-cell rings of 5 30-cell rings each. But these 150-cell rings do not have 5 30-cell rings around a common decagon axis, and 6 decagons each. Their axis is a 30-cell ring, not a decagon, and they contain only 3 decagons each. To construct them, on each of the four completely disjoint 30-cell rings, face-bond three more 30-cell rings to the exterior faces, making four stellated (&quot;bumpy&quot;) rings containing four 30-cell rings (120 cells) each. Collectively they contain 16 of the 20 30-cell rings: there are still four 30-cell ring &quot;holes&quot; left to fill in the 600-cell. To do that, fill some of the surface concavities of each 120-tetrahedron ring by wrapping a fifth 30-cell ring around its circumference, completely orthogonal to the axial 30-cell ring you started with. The result is four 150-cell tori, of 5 30-cell rings each, each having two completely orthogonal 30-cell ring axes, either of which can be seen as either an axis or a circumference: it is both.
On the icosahedron Hopf map,{{Efn|name=Hopf fibration base}} the four 30-cell rings lift from a star of four icosahedron faces (three faces edge-bonded around one). The fifth 30-cell ring lifts from a fifth face edge-bonded to the star, a sort of &quot;extra flap&quot; like the sixth square flap of the [[W:Cube#Orthogonal projections|net of a cube]] before you fold it up into a cube. It does not matter which of the six possible adjacent faces you choose as the flap, but the choice determines the choice for all four 150-cell rings. There are six choices because there are six decagonal fibrations; this is when you fix which fibration you are taking. Thus ''every'' 30-cell ring is the center core of a 150-cell ring.}}

==== Radial golden triangles ====
The 600-cell can be constructed radially from 720 [[W:Golden triangle (mathematics)|golden triangle]]s of edge lengths &lt;math&gt;1, 1, \phi^{-1}&lt;/math&gt; which meet at the center of the 4-polytope, each contributing two &lt;math&gt;\sqrt{1}&lt;/math&gt; radii and a &lt;math&gt;\phi^{-1}&lt;/math&gt; edge.
They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral &lt;math&gt;\phi^{-1}&lt;/math&gt; bases (the faces of the 600-cell).
These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular &lt;math&gt;\phi^{-1}&lt;/math&gt; tetrahedron bases (the cells of the 600-cell).

==== Characteristic orthoscheme ====
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;600-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;phi{-1} \approx 0.618&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;36°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{5}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;164°29′&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\pi-2\psi&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
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|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;22°15′20″&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3} - \eta&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;18°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{10}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;36°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{5}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;17°44′40″&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\eta - \tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4\phi^2}} \approx 0.535&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;22°15′20″&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3} - \eta&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4\phi^2}} \approx 0.309&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;18°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{10}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12\phi^2}} \approx 0.178&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;17°44′40″&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\eta - \tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac12\sqrt{2 + \phi} \approx 0.951&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^2}{3}} \approx 0.934&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}} \approx 0.926&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
|
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|-
!align=right|&lt;small&gt;&lt;math&gt;\eta&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|&lt;small&gt;37°44′40″&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\text{arc sec }4}{2}&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|}
Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls'').
Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center.
The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}
The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets.
It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]].
The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid &lt;small&gt;&lt;math&gt;\{p, q\}&lt;/math&gt;&lt;/small&gt; correspond to equatorial polygons of the truncation &lt;small&gt;&lt;math&gt;\{\tfrac{p}{q}\}&lt;/math&gt;&lt;/small&gt; and to ''equators'' of the simplicially subdivided spherical tessellation &lt;small&gt;&lt;math&gt;\{p, q\}&lt;/math&gt;&lt;/small&gt;. This &quot;[[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]&quot; is the arrangement of &lt;small&gt;&lt;math&gt;g = g_{p, q}&lt;/math&gt;&lt;/small&gt; right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called &quot;lines of symmetry&quot;, but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb &lt;small&gt;&lt;math&gt;\{p, q, r\}&lt;/math&gt;&lt;/small&gt; consists of the &lt;small&gt;&lt;math&gt;g = g_{p, q, r}&lt;/math&gt;&lt;/small&gt; tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope &lt;small&gt;&lt;math&gt;\{p, q, r\}&lt;/math&gt;&lt;/small&gt;. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a &quot;rotating ring of tetrahedra&quot; ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius.
The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center.
Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme.
The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 600-cell has unit radius and edge length &lt;small&gt;&lt;math&gt;\ell = \phi^{-1} \approx 0.618&lt;/math&gt;&lt;/small&gt;, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6\phi^2}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12\phi^2}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\tfrac12\sqrt{2 + \phi}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 600-cell).
The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4\phi^2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{\phi^4}{8}}&lt;/math&gt;&lt;/small&gt;, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.

==== Reflections ====
The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}}
Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation,{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; &quot;Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections.
Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections.&quot;}} so for example an ''n''-dimensional reflection is an (''n''+1)-dimensional half-turn.{{Sfn|Coxeter|1973|loc=§12-34|p=220}}

A full isoclinic revolution of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram&lt;sub&gt;2&lt;/sub&gt; geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}}
Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q&lt;sup&gt;2&lt;/sup&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q&lt;sup&gt;2&lt;/sup&gt;. By the same principle, we can view any QT or Q&lt;sup&gt;2&lt;/sup&gt; as an isoclinic (equi-angled) Q&lt;sup&gt;2&lt;/sup&gt; by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}}

==== Weyl orbits ====
Another construction method uses [[#Symmetries|quaternions]] and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits &lt;math&gt;O(\Lambda)=W(H_4)=I&lt;/math&gt; of order 120.{{Sfn|Koca, Al-Ajmi, &amp; Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4):
: O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
: O(1000) : V1
: O(0010) : V2
: O(0001) : V3

[[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]]

With quaternions &lt;math&gt;(p,q)&lt;/math&gt; where &lt;math&gt;\bar p&lt;/math&gt; is the conjugate of &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;[p,q]:r\rightarrow r'=prq&lt;/math&gt; and &lt;math&gt;[p,q]^*:r\rightarrow r''=p\bar rq&lt;/math&gt;, then the [[W:Coxeter group|Coxeter group]] &lt;math&gt;W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace &lt;/math&gt; is the symmetry group of the 600-cell and the [[120-cell|120-cell]] of order 14400.

Given &lt;math&gt;p \in T&lt;/math&gt; such that &lt;math&gt;\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p&lt;/math&gt; and &lt;math&gt;p^\dagger&lt;/math&gt; as an exchange of &lt;math&gt;-1/\varphi \leftrightarrow \varphi&lt;/math&gt; within &lt;math&gt;p&lt;/math&gt;, we can construct:

* the [[W:Snub 24-cell|snub 24-cell]] &lt;math&gt;S=\sum_{i=1}^4\oplus p^i T&lt;/math&gt;
* the 600-cell &lt;math&gt;I=T+S=\sum_{i=0}^4\oplus p^i T&lt;/math&gt; 
* the [[120-cell|120-cell]] &lt;math&gt;J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'&lt;/math&gt;

=== Rotations ===
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨&lt;sub&gt;4&lt;/sub&gt;.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one [[W:Completely orthogonal|completely orthogonal]] invariant plane rotates.
Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions).
Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles.
A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''.
Simple rotations are not commutative; left and right rotations (in general) reach different destinations.
The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles.
The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}}
Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}}
An '''isoclinic rotation''' is a different special case, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]].{{Efn|name=isoclinic geodesic}}
The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance.
(In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.)
All vertices are displaced to a vertex more than one edge-length away.{{Efn|name=isoclinic rotation to non-adjacent vertices}}
For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex &lt;math&gt;\sqrt{1}&lt;/math&gt; (60°) distant, moving &lt;math&gt;\sqrt{1/4} {{=}} 1/2&lt;/math&gt; (half the &lt;math&gt;\sqrt{1}&lt;/math&gt; overall displacement) in four orthogonal directions.|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.
A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points).
Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere).
Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}}
But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}}
Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once.
They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}}
They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.
These '''[[#Rotations on polygram isoclines|isoclines]]''' are geodesic 1-dimensional lines embedded in a 4-dimensional space.
On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in [[W:Chiral|chiral]] pairs as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}}
A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}}
The double loop is a true circle in four dimensions.{{Efn|name=one true 5𝝅 circle}}
Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]].
They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}}

The 600-cell is generated by [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes{{Efn|name=isoclinic invariant planes}} are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle.
A [[W:William Kingdon Clifford|Clifford]] displacement is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn||name=isoclinic 4-dimensional diagonal}}
Every plane that is Clifford parallel to one of the completely orthogonal planes is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane rotates sideways.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}}
''All'' central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).|name=Clifford displacement}}

==== Twenty-five 24-cells ====
There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}}
Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}

The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.{{Efn|name=Six orthogonal planes of the Cartesian basis}}

The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|𝜋|3}} to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|𝜋|3}}) isoclinically with respect to each other.
Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° ({{sfrac|2𝜋|3}}) apart.
In a unit-radius 4-polytope, vertices 120° apart are joined by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.|name=120° apart}}

The 120-vertex 600-cell has 60 long diameters: ''not just'' 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.{{Sfn|Waegell &amp; Aravind|2009|loc=§3. The 600-cell|pp=2-5}}
There ''are'' 5 disjoint 24-cells in the 600-cell, but not ''just'' 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[W:Pieter Hendrik Schoute|Schoute]] was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells.
The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells.
The rows and columns of the array are the only ten such partitions of the 600-cell.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|p=434}}}}

Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]].
The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane.
There are an infinite number of pairs of [[W:Completely orthogonal|completely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]].
The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}}
In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}}

Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon.
The two parallel decagons and the ten linking edges form a double helix ring.
Three decagons can also be parallel (decagons come in parallel [[W:Hopf fibration|fiber bundles]] of 12) and three of them may form a triple helix ring.
If the ring is cut and laid out flat in 3-space, it is a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long.
The three Clifford parallel decagons can be seen as the {{Background color|cyan}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]].
Each {{Background color|magenta}} edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}}
An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}}
Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the ''left'' of each 24-cell, and another 4 disjoint 24-cells to its ''right''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 600-cell}} so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}}
The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once.
Four paths are right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, and four are left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}|name=rotations to 8 disjoint 24-cells}}
The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells.

All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}}
Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and ''not'' disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared).
For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|16-cell]] (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).|name=isoclinic and not disjoint}}
Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}}
With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}}
Non-disjoint 24-cells are related by a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane.
They are also related by an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|simple rotation]] which will take any vertex ''directly'' to any other vertex, also moving most or all of the other vertices but leaving at most 6 other vertices fixed (the vertices that the fixed central plane intersects).
The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[W:Digon|digon]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively.
Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|𝜋|5}} of the digon central plane completely orthogonal to their common hexagonal central plane.
In this simple rotation, the hexagon does not move.
The two ''non-disjoint'' 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane ''does'' move.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}}

There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}}
As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} through 4-space.
Note however that in a ''discrete'' decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain ''no'' points.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}}
''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|fibration of 12 Clifford parallel ''decagonal'' invariant planes]].
(There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.){{Efn|name=rotations to 8 disjoint 24-cells}}
''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart.
However, [[#Icosahedra|recall]] that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart.
The hexagons have their own [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}}
(There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}}
Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates ''around'' the common hexagonal plane.
The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}
In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is ''not'' fixed: it rotates (non-invariantly) through {{sfrac|𝜋|5}}.
The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}}

On the other hand, each of the 10 sets of five ''disjoint'' 24-cells is Clifford parallel because its corresponding great ''hexagons'' are Clifford parallel.
(24-cells do not have great decagons.)
The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel [[24-cell#Geodesics|geodesics]], each set of which covers all 24 vertices of the 24-cell.
The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel [[#Geodesics|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|hexagonal fibration]].
Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell.
Similarly, the corresponding great ''squares'' of disjoint 24-cells are Clifford parallel.

==== Rotations on polygram isoclines ====
The regular convex 4-polytopes each have their characteristic kind of right (and left) [[W:Isoclinic rotation|isoclinic rotation]], corresponding to their characteristic kind of discrete [[W:Hopf fibration|Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[W:Completely orthogonal|completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.''{{Efn|Consider the statement: ''In one full revolution of an isoclinic rotation, every point in the space loops exactly once through its great circle Hopf fiber.'' It can be found in the literature, expressed in the mathematical language of the Hopf fibration,{{Sfn|Kim|Rote|2016|loc=
8 The Construction of Hopf Fibrations|pp=12-16|ps=; see Theorem 13.}} but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting sideways, rotating in parallel with the completely orthogonal great circle.{{Efn|name=isoclinic geodesic}} With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle.{{Efn|name=helical geodesic}} Each helical isocline is itself a kind of circle, but it is not a planar great circle of the [[W:Hopf fibration|Hopf fibration]]: it is a special kind of geodesic circle whose circumference is greater than 2𝝅''r'', and it is not pictured explicitly at all by the plain statement we are trying to visualize.{{Efn|name=isocline circumference.}} We cannot easily visualize this statement about the Hopf great circles in a stationary reference frame. The statement does ''not'' simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle ''as every great circle itself is moving orthogonally'', flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous ''twisting'' rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different rotating wheels, because Clifford parallel circles are not parallel in the ordinary sense, and every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once ''in a full isoclinic revolution, every vertex moves more than 360 degrees,'' as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique Hopf fibration,{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the [[24-cell#Isoclinic rotations|isoclinic rotation of a great hexagon fibration of the 24-cell]], each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle);{{Efn|name=4𝝅 rotation}} but in the [[#Decagons and pentadecagrams|isoclinic rotation of a great decagon fibration of the 600-cell]], each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).}} The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[W:Chain mail|chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell.

Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[W:Skew polygon|skew]] ''[[W:Polygram (geometry)|polygrams]]'' (which trace the paths on the [[W:Clifford torus|Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[W:Fiber bundle|fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}}
Since isoclinic rotations are [[W:Chiral|chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}}
All the fiber bundles are aspects of the same discrete [[W:Hopf fibration|Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations.

Cell rings are another expression of the Hopf fibration.
Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating.
In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}}
The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells.
The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets.

Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[W:Loom|loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers.
A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[W:Warp and woof|warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes.
Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}}

In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§1. Introduction|ps=; &quot;This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization.&quot;{{Efn|name=double rotation}}}}
The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[W:Chiral|chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|Clifford parallel cell rings]] may or may not be [[W:Chiral|chiral]] objects, depending upon whether the 4-polytope's cells have opposing faces or not.
The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise.
Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}}
The characteristic cell rings of the 8-cell tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no [[W:Torsion of a curve|torsion]]).
Pairs of left-handed and right-handed isoclines run through cell rings of this kind. The left and right isoclines are enantiomorphously congruent (mirror images) of each other.
Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}}
Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively.
Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once.
It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers.

Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle.
There are two ways they can do this: by both rotating in the &quot;same&quot; direction, or by rotating in &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).
The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}
Left and right isoclines are different paths that go to different places.
In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers.

A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation.
Each moving vertex travels along an isocline contained within a (moving) cell ring.
While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}}
The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations.
Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines.
The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}}

A [[24-cell#Simple rotations|simple rotation]] is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane. (The 600-cell has four orthogonal [[#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.)
In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration.

An [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices{{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex.
In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is the opposite vertex of a neighboring cell.
In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron.
|name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind).
A left-right pair of isoclinic rotations constitutes a discrete fibration.
All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by ''two'' equal angles and lying in different hyperplanes.{{Efn|name=two angles between central planes}}
The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[W:Geodesic|geodesic]].

==== Decagons and pentadecagrams ====
The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines.

The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}}
12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[W:Hopf fibration|Hopf fibration]].
There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}}
The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons.
The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}}
A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]].
A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration).
The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether.
Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices.

In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}}
In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon.
In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a &lt;s&gt;[[W:Hexagram|hexagram]]&lt;/s&gt;: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; chords of the hexagon instead of &lt;math&gt;\sqrt{1}&lt;/math&gt; hexagon edges.{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle. The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) although all such circles of the same circumference are directly or enantiomorphously congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=4-polytopes ordered by size and complexity}}
In the &lt;math&gt;\sqrt{1}&lt;/math&gt; [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply &lt;math&gt;\sqrt{3}&lt;/math&gt; chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic &lt;s&gt;{6/2} hexagram&lt;/s&gt; rotation]] both rotate circles of 6 vertices.
The &lt;s&gt;hexagram&lt;/s&gt; isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}}
The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of &lt;s&gt;hexagrams&lt;/s&gt;]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}}
In the &lt;math&gt;\sqrt{2}&lt;/math&gt; [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its &lt;math&gt;\sqrt{2}&lt;/math&gt; edges and its &lt;math&gt;\sqrt{4}&lt;/math&gt; diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[W:Winding number|winds]] ''three'' times around the 3-sphere.
The invariant central plane completely orthogonal to each {4} great square is another {4} great square &lt;math&gt;\sqrt{4}&lt;/math&gt; distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares.
The 16-cell's [[W:Dural polytope|dual polytope]] the [[W:8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). 
In the 8-cell this is a rotation of &lt;math&gt;\sqrt{1}&lt;/math&gt; × &lt;math&gt;\sqrt{3}&lt;/math&gt; great rectangles, and also a rotation of &lt;math&gt;\sqrt{4}&lt;/math&gt; axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]].
In the &lt;math&gt;\phi^{-1}&lt;/math&gt; [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are &lt;math&gt;\sqrt{1}&lt;/math&gt; hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}}
The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices.
The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices.
The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain {2} great [[W:Digon|digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell.
The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis.
The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=4-polytopes ordered by size and complexity}} nested like [[W:Russian dolls|Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} 
This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}}

In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it.
Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}}
The two non-adjacent vertices are linked by a &lt;math&gt;\sqrt{1}&lt;/math&gt; chord of the isocline which is a great hexagon edge (a 24-cell edge).
The &lt;math&gt;\sqrt{1}&lt;/math&gt; chords of the 30-cell ring (without the &lt;math&gt;\phi^{-1}&lt;/math&gt; 600-cell edges) form a skew [[W:Triacontagram|triacontagram]]&lt;sub&gt;{30/4}=2{15/2}&lt;/sub&gt; which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[W:Pentadecagram|pentadecagram]]&lt;sub&gt;2&lt;/sub&gt; isoclines.
Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers.
Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[W:Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors.
Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]], '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. (Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.)
Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[#Boerdijk–Coxeter helix rings|600-cell]].
Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]].
Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as part of left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}}
Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}}
The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations.

At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end.
Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).}}
Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation.
Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines.
There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell.
Each distinct isocline belongs to 5 fiber bundles.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=4|Three sets of 30-cell ring chords from the same [[W:Orthogonal projection|orthogonal projection]] viewpoint
|-
![[W:Pentadecagon#Pentadecagram|Pentadecagram {15/2}]]
![[W:Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]]
![[W:Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]]
|-
|colspan=2 align=center|All edges are [[W:Pentadecagram|pentadecagram]] isocline chords of length &lt;math&gt;\sqrt{1}&lt;/math&gt;, which are also [[24-cell#Great hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell.
|colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length &lt;math&gt;\sqrt{3 - \phi} \approx 1.176&lt;/math&gt;. 
|-
|[[File:Regular_star_polygon_15-2.svg|200px]]
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_6(5,1).svg|200px]]
|-
|valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The &lt;math&gt;\sqrt{1}&lt;/math&gt; chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart.
|valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The &lt;math&gt;\sqrt{1}&lt;/math&gt; chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra.
|valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly or enantiomorphously congruent.
|- 
|colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. 
|}

Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings. The right and left isoclines in each 3-cell ring are enantiomorphously congruent (mirror images).{{Efn|The chord-path of an isocline may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}
The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit.
The double loop is entirely contained within a single [[#Boerdijk–Coxeter helix rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}}
Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell.
Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart.
Thus each cell has two helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points.
Globally these two helices are a single connected circle of ''both'' chiralities,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly or enantiomorphously congruent.{{Efn|name=isocline circumference.}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[W:Torsion of a curve|torsion]].{{Efn|name=Sadoc frustration}}
An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}}
The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one &lt;math&gt;\phi^{-1}&lt;/math&gt; edge-length apart).
The 30 chords joining the isocline's 30 vertices are &lt;math&gt;\sqrt{1}&lt;/math&gt; hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell &lt;math&gt;\phi^{-1}&lt;/math&gt; edges apart on a decagon great circle.
{{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram&lt;sub&gt;2&lt;/sub&gt; geodesic]] is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right).
The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions.
The Möbius helix is a geodesic &quot;straight line&quot; or ''[[#Decagons and pentadecagrams|isocline]]''.
The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon.
The Petrie triacontagon has &lt;math&gt;\phi^{-1}&lt;/math&gt; edges; the isoclinic pentadecagram&lt;sub&gt;2&lt;/sub&gt; has &lt;math&gt;\sqrt{1}&lt;/math&gt; edges which join vertices which are two &lt;math\phi^{-1}&lt;/math&gt; edges apart.
Each &lt;math&gt;\sqrt{1}&lt;/math&gt; edge belongs to a different [[#Hexagons|great hexagon]], and successive &lt;math&gt;\sqrt{1}&lt;/math&gt; edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.|name=double threaded}}
These isocline chords are both hexa''gon'' edges and penta''gram'' edges.

The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other.
Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[W:Chessboard|chessboard]].{{Efn|name=isoclinic chessboard}}
In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring.

==== Hexagons and &lt;s&gt;hexagrams&lt;/s&gt; ====
[[File:Regular_star_figure_2(10,3).svg|thumb|[[W:Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines.

Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon.
The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black &lt;math&gt;\sqrt{3}&lt;/math&gt; [[24-cell#Great triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring.
The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 &lt;math&gt;\sqrt{3 - \phi}&lt;/math&gt; chords of each isocline form a skew [[W:Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[W:Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[W:Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]].

Notice the relation between the [[24-cell#Helical dodecagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on &lt;s&gt;hexagram&lt;/s&gt; isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline because the 600-cell contains multiple 24-cells, and the 600-cell's &lt;math&gt;\sqrt{3 - \phi}&lt;/math&gt; isocline chord is shorter than the 24-cell's &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on &lt;s&gt;[[24-cell#Helical dodecagrams and their isoclines|hexagrams]]&lt;/s&gt;, while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their directly or enantiomorphously congruent isoclines are all a geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference 2𝝅''r''; simple rotations take place on these isoclines.  Double rotations have isoclines of more than 2𝝅''r'' circumference, because their circle does not close in a single revolution. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic (equi-angled) double rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅''r'' circumference. The 600-cell edge-rotates on isoclines of 5𝝅''r'' circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[W:20-gon|icosagram]] is a compound of the 24-cell's &lt;s&gt;helical {6/2} hexagram&lt;/s&gt;, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}}

==== Squares and octagrams ====
[[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[W:24-gon#Related polygons|{24/5} 24-gram]], with &lt;math&gt;\phi&lt;/math&gt; edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell (&lt;math&gt;\sqrt{1}&lt;/math&gt; edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines.

Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines.

The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 &lt;math&gt;\phi&lt;/math&gt; chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one &lt;math&gt;\sqrt{1}&lt;/math&gt; chord apart, and 5 &lt;math&gt;\phi&lt;/math&gt; chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell.

Notice the relations between the [[16-cell#Helical construction|16-cell's rotation in just 2 completely orthogonal great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their directly or enantiomorphously congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the &lt;math&gt;\sqrt{4}&lt;/math&gt; diameter. In the 600-cell vertices are closer together, and its &lt;math&gt;\phi&lt;/math&gt; chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference.{{Efn|name=isocline circumference.}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 600-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

&lt;math&gt;\begin{bmatrix}\begin{matrix}120 &amp; 12 &amp; 30 &amp; 20 \\ 2 &amp; 720 &amp; 5 &amp; 5 \\ 3 &amp; 3 &amp; 1200 &amp; 2 \\ 4 &amp; 6 &amp; 4 &amp; 600 \end{matrix}\end{bmatrix}&lt;/math&gt;

Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal.
{| class=wikitable
!H&lt;sub&gt;4&lt;/sub&gt;||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}
! [[W:k-face|''k''-face]]||f&lt;sub&gt;''k''&lt;/sub&gt;||f&lt;sub&gt;0&lt;/sub&gt; || f&lt;sub&gt;1&lt;/sub&gt;||f&lt;sub&gt;2&lt;/sub&gt;||f&lt;sub&gt;3&lt;/sub&gt;||[[W:Vertex figure|''k''-fig]]
!Notes
|- align=right
|H&lt;sub&gt;3&lt;/sub&gt; || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|5|node}} ||( )
!f&lt;sub&gt;0&lt;/sub&gt;
|| 120 ||  12 ||   30 ||  20 ||[[W:icosahedron|{3,5}]] || H&lt;sub&gt;4&lt;/sub&gt;/H&lt;sub&gt;3&lt;/sub&gt; = 14400/120 = 120
|- align=right
|A&lt;sub&gt;1&lt;/sub&gt;H&lt;sub&gt;2&lt;/sub&gt; ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|5|node}} ||{ }
!f&lt;sub&gt;1&lt;/sub&gt;
||   2 || 720 ||    5 ||   5 || [[W:pentagon|{5}]] || H&lt;sub&gt;4&lt;/sub&gt;/H&lt;sub&gt;2&lt;/sub&gt;A&lt;sub&gt;1&lt;/sub&gt; = 14400/10/2 = 720
|- align=right
|A&lt;sub&gt;2&lt;/sub&gt;A&lt;sub&gt;1&lt;/sub&gt; ||{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node}} ||[[W:equilateral triangle|{3}]]
!f&lt;sub&gt;2&lt;/sub&gt;
||   3 ||   3 || 1200 ||   2 || { } || H&lt;sub&gt;4&lt;/sub&gt;/A&lt;sub&gt;2&lt;/sub&gt;A&lt;sub&gt;1&lt;/sub&gt; = 14400/6/2 = 1200
|- align=right
|A&lt;sub&gt;3&lt;/sub&gt; ||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}} ||[[W:tetrahedron|{3,3}]]
!f&lt;sub&gt;3&lt;/sub&gt;
||   4 ||   6 ||    4 || 600|| ( ) || H&lt;sub&gt;4&lt;/sub&gt;/A&lt;sub&gt;3&lt;/sub&gt; = 14400/24 = 600
|}

== Symmetries ==
The [[W:Icosian|icosian]]s are a specific set of Hamiltonian [[W:Quaternion|quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}}
The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[W:Rational number|rational number]]s.{{Sfn|Steinbach|1997|p=24}}
The finite sums of the 120 [[W:Icosian#Unit icosians|unit icosians]] are called the [[W:Icosian#Icosian ring|icosian ring]].

When interpreted as quaternions,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate.
[[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]].
[[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century.
Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}}
Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the 120 vertices of the 600-cell form a [[W:group (mathematics)|group]] under quaternionic multiplication.
This group is often called the [[W:Binary icosahedral group|binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[W:Icosahedral group|icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}}
It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[W:Invariant subgroup|invariant subgroup]], namely as the subgroup ''2I&lt;sub&gt;L&lt;/sub&gt;'' of quaternion left-multiplications and as the subgroup ''2I&lt;sub&gt;R&lt;/sub&gt;'' of quaternion right-multiplications.
Each rotational symmetry of the 600-cell is generated by specific elements of ''2I&lt;sub&gt;L&lt;/sub&gt;'' and ''2I&lt;sub&gt;R&lt;/sub&gt;''; the pair of opposite elements generate the same element of ''RSG''.
The [[W:Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''.
We have the isomorphism ''RSG ≅ (2I&lt;sub&gt;L&lt;/sub&gt; × 2I&lt;sub&gt;R&lt;/sub&gt;) / {Id, -Id}''.
The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200.
The [[W:Quaternion algebra|quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|1994|p=1|loc=&quot;''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions.&quot;}}

The binary icosahedral group is [[W:Isomorphic|isomorphic]] to [[W:special linear group|SL(2,5)]].

The full [[W:Symmetry group|symmetry group]] of the 600-cell is the [[W:H4 (mathematics)|Coxeter group H&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=§2 The Labeling of H&lt;sub&gt;4&lt;/sub&gt;}}
This is a [[W:Group (mathematics)|group]] of order 14400.
It consists of 7200 [[W:Rotation (mathematics)|rotations]] and 7200 rotation-reflections.
The rotations form an [[W:Invariant subgroup|invariant subgroup]] of the full symmetry group.
The rotational symmetry group was first described by S.L. van Oss.{{Sfn|van Oss|1899||pp=1-18}}
The H&lt;sub&gt;4&lt;/sub&gt; group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; &quot;[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'.
In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly.
Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes....
This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.&quot;}}

== Visualization ==
The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,{{Efn||name=tetrahedral cell adjacency}} and the fact that the tetrahedron has no opposing faces or vertices.{{Efn|name=directly congruent versus twisted cell rings}}  One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} which with some effort can be seen in most of the below perspective projections.

=== 2D projections ===
The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]].

{| class=&quot;wikitable&quot; width=600
|+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
|- align=center
!H&lt;sub&gt;4&lt;/sub&gt;
! -
!F&lt;sub&gt;4&lt;/sub&gt;
|- align=center
|[[File:600-cell graph H4.svg|200px]]&lt;br&gt;[30]&lt;br&gt;(Red=1)
|[[File:600-cell t0 p20.svg|200px]]&lt;br&gt;[20]&lt;br&gt;(Red=1)
|[[File:600-cell t0 F4.svg|200px]]&lt;br&gt;[12]&lt;br&gt;(Red=1)
|- align=center
!H&lt;sub&gt;3&lt;/sub&gt;
!A&lt;sub&gt;2&lt;/sub&gt; / B&lt;sub&gt;3&lt;/sub&gt; / D&lt;sub&gt;4&lt;/sub&gt;
!A&lt;sub&gt;3&lt;/sub&gt; / B&lt;sub&gt;2&lt;/sub&gt;
|- align=center
|[[File:600-cell t0 H3.svg|200px]]&lt;br&gt;[10]&lt;br&gt;(Red=1,orange=5,yellow=10)
|[[File:600-cell t0 A2.svg|200px]]&lt;br&gt;[6]&lt;br&gt;(Red=1,orange=3,yellow=6)
|[[File:600-cell t0.svg|200px]]&lt;br&gt;[4]&lt;br&gt;(Red=1,orange=2,yellow=4)
|}

=== 3D projections ===
A three-dimensional model of the 600-cell, in the collection of the [[W:Institut Henri Poincaré|Institut Henri Poincaré]], was photographed in 1934–1935 by [[W:Man Ray|Man Ray]], and formed part of two of his later &quot;Shakesperean Equation&quot; paintings.&lt;ref&gt;{{citation|title=Man Ray Human Equations: A journey from mathematics to Shakespeare|publisher=Hatje Cantz|editor1-first=Wendy A.|editor1-last=Grossman|editor2-first=Edouard|editor2-last=Sebline|year=2015}}. See in particular ''mathematical object mo-6.2'', p.&amp;nbsp;58; ''Antony and Cleopatra'', SE-6, p.&amp;nbsp;59; ''mathematical object mo-9'', p.&amp;nbsp;64; ''Merchant of Venice'', SE-9, p.&amp;nbsp;65, and &quot;The Hexacosichoron&quot;, Philip Ordning, p.&amp;nbsp;96.&lt;/ref&gt;

{| class=wikitable
!colspan=2|Vertex-first projection
|-
|[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]]
|This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
* The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown.
* The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow.
* The remaining cells are rendered in edge-outline.
* Cells facing away from the 4D viewpoint (those lying on the &quot;far side&quot; of the 600-cell) have been culled, to reduce visual clutter in the final image.
|-
!colspan=2|Cell-first projection
|-
|[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]]
|This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
* The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image.
* The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow.
* The remaining cells are rendered in edge-outline.
* Cells facing away from the 4D viewpoint have been culled for clarity.

This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.
|}

=== Animations===

{| class=wikitable width=540
!colspan=1|Coxeter section views
|-
|align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]&lt;br&gt;Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls.
|}

== Diminished 600-cells ==
The [[W:Snub 24-cell|snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell|24-cell]] and taking the [[W:Convex hull|convex hull]] of the remaining vertices.{{Sfn|Dechant|2021|pp=22-24|loc=§8. Snub 24-cell}} This process is a ''[[W:Diminishment (geometry)|diminishing]]'' of the 600-cell.

The [[W:Grand antiprism|grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H&lt;sub&gt;2&lt;/sub&gt; × H&lt;sub&gt;2&lt;/sub&gt;}}

A bi-24-diminished 600-cell, with all [[W:Tridiminished icosahedron|tridiminished icosahedron]] cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells.

There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.&lt;ref&gt;{{Cite journal|last1=Sikiric|first1=Mathieu|last2=Myrvold|first2=Wendy|date=2007|title=The special cuts of 600-cell|journal=Beiträge zur Algebra und Geometrie|volume=49|issue=1|arxiv=0708.3443}}&lt;/ref&gt;
{| class=&quot;wikitable collapsible&quot;
!colspan=12|Diminished 600-cells
|-
!Name
!Tri-24-diminished 600-cell
!Bi-24-diminished 600-cell
![[W:Snub 24-cell|Snub 24-cell]]&lt;br&gt;(24-diminished 600-cell)
![[W:Grand antiprism|Grand antiprism]]&lt;br&gt;(20-diminished 600-cell)
!600-cell
|- align=center
!Vertices
|48
|72
|96
|100
|120
|- align=center
!Vertex figure&lt;br&gt;(Symmetry)
|[[File:Dual tridiminished icosahedron.png|120px]]&lt;br&gt;dual of tridiminished icosahedron&lt;br&gt;([3], order 6)
|[[File:Biicositetradiminished 600-cell vertex figure.png|120px]]&lt;br&gt;[[W:Hexahedron|tetragonal antiwedge]]&lt;br&gt;([2]&lt;sup&gt;+&lt;/sup&gt;, order 2)
|[[File:Snub 24-cell verf.png|120px]]&lt;br&gt;[[W:tridiminished icosahedron|tridiminished icosahedron]]&lt;br&gt;([3], order 6)
|[[File:Grand antiprism verf.png|120px]]&lt;br&gt;[[W:Edge-contracted icosahedron|bidiminished icosahedron]]&lt;br&gt;([2], order 4)
|[[File:600-cell verf.svg|120px]]&lt;br&gt;[[W:Icosahedron|icosahedron]]&lt;br&gt;([5,3], order 120)
|- align=center
!Symmetry
|colspan=2|Order 144 (48×3 or 72×2)
|[3&lt;sup&gt;+&lt;/sup&gt;,4,3]&lt;br&gt;Order 576 (96×6)
|[10,2&lt;sup&gt;+&lt;/sup&gt;,10]&lt;br&gt;Order 400 (100×4)
|[5,3,3]&lt;br&gt;Order 14400 (120×120)
|- align=center
!Net
|[[File:Triicositetradiminished hexacosichoron net.png|100px]]
|[[File:Biicositetradiminished hexacosichoron net.png|100px]]
|[[File:Snub 24-cell-net.png|100px]]
|[[File:Grand antiprism net.png|100px]]
|[[File:600-cell net.png|100px]]
|- align=center
!Ortho&lt;br&gt;H&lt;sub&gt;4&lt;/sub&gt; plane
|[[File:Tridiminished 600-cell H4 Coxeter plane.svg|120px]]
|[[File:bidex ortho-30-gon.png|120px]]
|[[File:Snub 24-cell ortho30-gon.png|120px]]
|[[File:Grand antiprism ortho-30-gon.png|120px]]
|[[File:600-cell graph H4.svg|120px]]
|- align=center
!Ortho&lt;br&gt;F&lt;sub&gt;4&lt;/sub&gt; plane
|[[File:Tridiminished 600-cell F4 Coxeter plane.svg|120px]]
|[[File:Bidex ortho 12-gon.png|120px]]
|[[File:24-cell h01 F4.svg|120px]]
|[[File:GrandAntiPrism-2D-F4.svg|120px]]
|[[File:600-cell t0 F4.svg|120px]]
|}

== Related polytopes and honeycombs ==
The 600-cell is one of 15 regular and uniform polytopes with the same H&lt;sub&gt;4&lt;/sub&gt; symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020}}
{{H4_family}}

It is similar to three [[W:Regular 4-polytope|regular 4-polytope]]s: the [[5-cell|5-cell]] {3,3,3}, [[16-cell|16-cell]] {3,3,4} of Euclidean 4-space, and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have [[W:Tetrahedron|tetrahedral]] cells.
{{Tetrahedral cell tessellations}}

This 4-polytope is a part of a sequence of 4-polytope and honeycombs with [[W:Icosahedron|icosahedron]] vertex figures:
{{Icosahedral vertex figure tessellations}}

The [[W:regular complex polytope|regular complex polygons]] &lt;sub&gt;3&lt;/sub&gt;{5}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|5|3node}} and &lt;sub&gt;5&lt;/sub&gt;{3}&lt;sub&gt;5&lt;/sub&gt;, {{Coxeter–Dynkin diagram|5node_1|3|5node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; have a real representation as ''600-cell'' in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has [[W:Complex reflection group|complex reflection group]] &lt;sub&gt;3&lt;/sub&gt;[5]&lt;sub&gt;3&lt;/sub&gt;, order 360, and the second has symmetry &lt;sub&gt;5&lt;/sub&gt;[3]&lt;sub&gt;5&lt;/sub&gt;, order 600.{{Sfn|Coxeter|1991|pp=48-49}}

{| class=&quot;wikitable collapsed collapsible&quot;
!colspan=3| Regular complex polytope in orthogonal projection of H&lt;sub&gt;4&lt;/sub&gt; Coxeter plane{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
|- align=center
|[[File:600-cell graph H4.svg|240px]]&lt;br&gt;{3,3,5}&lt;br&gt;Order 14400
|[[File:Complex polygon 3-5-3.png|240px]]&lt;br&gt;&lt;sub&gt;3&lt;/sub&gt;{5}&lt;sub&gt;3&lt;/sub&gt;&lt;br&gt;Order 360
|[[File:Complex polygon 5-3-5.png|240px]]&lt;br&gt;&lt;sub&gt;5&lt;/sub&gt;{3}&lt;sub&gt;5&lt;/sub&gt;&lt;br&gt;Order 600
|}

== See also ==
* [[W:600-cell|Wikipedia:600-cell]], the article this article is an expanded version of
* [[24-cell|24-cell]], the predecessor 4-polytope on which the 600-cell is based
* [[120-cell|120-cell]], the dual 4-polytope to the 600-cell, and its successor
* [[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]]
* [[W:Regular 4-polytope|Regular 4-polytope]]
* [[W:Polytope|Polytope]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
* {{Citation | last=Schläfli | first=Ludwig | author-link=W:Ludwig Schläfli |editor-first=Arthur | editor-last=Cayley | editor-link=W:Arthur Cayley | title=An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface | url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002 | year=1858 | journal=Quarterly Journal of Pure and Applied Mathematics | volume=2 | pages=55–65, 110–120 }}
{{Regular convex 4-polytopes Refs|wiki=W:}}


* {{Cite journal | first1=F. | last1=Buekenhout | first2=M. | last2=Parker | title=The number of nets of the regular convex polytopes in dimension &lt;= 4 | journal=[[W:Discrete Mathematics (journal)|Discrete Mathematics]] | volume=186 | issue=1–3 | date=15 May 1998 | pages=69–94| doi=10.1016/S0012-365X(97)00225-2 | doi-access=free | ref={{SfnRef|Buekenhout &amp; Parker|1998}} }}
* {{cite journal | last1 = Itoh | first1 = Jin-ichi | last2 = Nara | first2 = Chie | doi = 10.1007/s00022-021-00575-6 | doi-access = free | issue = 13 | journal = [[W:Journal of Geometry|Journal of Geometry]] | title = Continuous flattening of the 2-dimensional skeleton of a regular 24-cell | volume = 112 | year = 2021 | ref={{SfnRef|Itoh &amp; Nara|2021}} }}
* [http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation  [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }}
* {{Cite journal | last=Oss | first=Salomon Levi van | title=Das regelmässige Sechshundertzell und seine selbstdeckenden Bewegungen | journal=Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 (Afdeeling Natuurkunde) | volume=7 | issue=1 | pages=1–18 | place=Amsterdam | year=1899 | url=https://books.google.com/books?id=AfQ3AQAAMAAJ&amp;pg=PA3 | ref={{SfnRef|van Oss|1899}} }}
{{Refend}}

== External links ==
* [https://bendwavy.org/klitzing/incmats/ex.htm ex], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Hexacosichoron Hexacosichoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Hydrochoron Hydrochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/600-cell The 600-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]

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      <text bytes="243943" sha1="m7v7k6pd0p7zgn99u02ihdothiin9rv" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="243949" sha1="e72rv0hjqkel8csiewydj5x9sesmz2q" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
      <sha1>e72rv0hjqkel8csiewydj5x9sesmz2q</sha1>
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      <text bytes="243655" sha1="8l1biijrew6lazo0jd6m0vovsjyk9hw" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}}

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="243326" sha1="ny22kwns3umzvmvdtwgpz3n9zzv209z" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
      <sha1>ny22kwns3umzvmvdtwgpz3n9zzv209z</sha1>
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        <username>Dc.samizdat</username>
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      <comment>/* Chiral symmetry operations */ hexagrams should be replaced by {12/5} dodecagrams -- notice the struck-thru text -- some of the star polygon images in this table are the wrong ones</comment>
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      <text bytes="243333" sha1="51lwrv95t020thcf4ub7uui40t2l5iy" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew hexagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|&lt;s&gt;hexagrams&lt;/s&gt;]], each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="243156" sha1="9zn4li63hspcuj8m5peas8dhl01kq2m" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew dodecagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew dodecagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|&lt;s&gt;hexagrams&lt;/s&gt;]], each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="243152" sha1="9gqxfqtd9svipfklxl6zbivztlgiw4y" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew dodecagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew dodecagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;.{{Efn|name=skew dodecagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Hexagram|dodagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|&lt;s&gt;hexagrams&lt;/s&gt;]], each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="242667" sha1="190vp5k8grbnhjaro0s3sksoojzs6gt" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew dodecagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew dodecagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;.{{Efn|name=skew dodecagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white isoclinic dodecagram geodesics run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains two such isoclinic skew dodecagrams, one black and one white.{{Efn|name=dodecagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|&lt;s&gt;hexagrams&lt;/s&gt;]], each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="242411" sha1="nvk8rqr1rtvh5kmj2c5zo9jkqg7e4ih" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew dodecagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew dodecagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;.{{Efn|name=skew dodecagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|&lt;s&gt;hexagrams&lt;/s&gt;]], each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="240894" sha1="p8z1759p7ps4gj8uqo10zr1ddogitca" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell &amp; Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Great triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell &amp; Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

{| class=wikitable
|- align=center
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f||style=&quot;background-color:#FFE119;&quot;|c
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||12||6
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||3||3
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f||3||3||style=&quot;background-color:#f0FFE0&quot;|'''96'''||2
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c||6||12||8||style=&quot;background-color:#f0FFE0&quot;|'''24'''
|}

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

In the [[W:uniform 4-polytope|uniform]] D&lt;sub&gt;4&lt;/sub&gt; construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.&lt;ref&gt;[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]&lt;/ref&gt; The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells.

{| class=wikitable
|\||style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E6194B;&quot;|e||style=&quot;background-color:#3CB44B;&quot;|f1||style=&quot;background-color:#3CB44B;&quot;|f2||style=&quot;background-color:#3CB44B;&quot;|f3||style=&quot;background-color:#FFE119;&quot;|c1||style=&quot;background-color:#FFE119;&quot;|c2||style=&quot;background-color:#FFE119;&quot;|c3
|- align=right
|align=left style=&quot;background-color:#808080;&quot;|v||style=&quot;background-color:#E0F0FF&quot;|'''24'''||8||4||4||4||2||2||2
|- align=right
|align=left style=&quot;background-color:#E6194B;&quot;|e||2||style=&quot;background-color:#f0FFE0&quot;|'''96'''||1||1||1||1||1||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f1||3||3||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||1||1||0
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f2||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||style=&quot;background-color:#f0FFE0&quot;|*||1||0||1
|- align=right
|align=left style=&quot;background-color:#3CB44B;&quot;|f3||3||3||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''32'''||0||1||1
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c1||6||12||4||4||0||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c2||6||12||4||0||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''||style=&quot;background-color:#f0FFE0&quot;|*
|- align=right
|align=left style=&quot;background-color:#FFE119;&quot;|c3||6||12||0||4||4||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|*||style=&quot;background-color:#f0FFE0&quot;|'''8'''
|}

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia &amp; Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia &amp; Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew dodecagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram&lt;sub&gt;5&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges.{{Efn|name=skew dodecagram}} All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986&lt;ref&gt;{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman &amp; Weinberg|1987}}}}&lt;/ref&gt; to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference &lt;math&gt;8\pi&lt;/math&gt;, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram&lt;sub&gt;5&lt;/sub&gt;.{{Efn|name=skew dodecagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}}  

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their  isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell &amp; Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell &amp; Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical dodecagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]&lt;sub&gt;5&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell &amp; Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}}

Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram&lt;sub&gt;5&lt;/sub&gt; path. &lt;s&gt;Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew dodecagram&lt;sub&gt;5&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram&lt;sub&gt;5&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.&lt;/s&gt;

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram&lt;sub&gt;5&lt;/sub&gt; geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|name=dodecagram}}&lt;br&gt;[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|&lt;s&gt;hexagrams&lt;/s&gt;]], each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}


== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}
&lt;br&gt;
* {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}}
* {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh &amp; Nara|2021}}}}
{{Refend}}

==External links==
* [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes]
* [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki]
* [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space]
* [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space]
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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      <text bytes="30443" sha1="5e9c6bb7etgc557dub6l2fdn38pjbj6" xml:space="preserve">{{title|Mobile phone use motivation:&lt;br&gt;What are the motivations for mobile phone use?}}
__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}

'''Imagine this ...'''

You are sitting on a crowded train after a long day. A young boy nearby laughs at a meme on Instagram. A woman beside him scrolls through her messages. Across the aisle, a student types reminders into a calendar app. Next to you, a man endlessly scrolls through social media updates. What draws each of these people to their phone? Is it comfort, connection, distraction, or control? These are everyday actions, yet behind them lie hidden motivations that shape our relationship with smartphones.
{{RoundBoxBottom}}

[[wikipedia:Mobile_phone|Mobile phones]] are central to modern life, influencing how people work, connect, and cope with [[wikipedia:Stress_(biology)|stress]]. Their use is driven by complex psychological needs that affect daily [[wikipedia:Behavior|behaviour]] and [[wikipedia:Well-being|wellbeing]]. Understanding these [[wikipedia:Motivation|motivations]] reveals why people engage with phones in unique ways, and why these behaviours can bring both benefits and challenges. [[wikipedia:Smartphone|Smartphones]] enhance productivity, entertainment, and social bonds. Yet, excessive or compulsive use can lead to [[wikipedia:Procrastination|procrastination]], [[wikipedia:Anxiety|anxiety]], and emotional [[wikipedia:Avoidance_coping|avoidance]]. This chapter explores how motivations shape phone use and offers strategies for healthier, intentional smartphone use that supports wellbeing.

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What motivates people to use smartphones in their daily lives?
* What psychological theories explain these motivations?
* How do different motivations impact wellbeing in positive and negative ways?
* What strategies can encourage mindful and beneficial phone use?
{{RoundBoxBottom}}

== Key motivations for mobile phone use ==
People use mobile phones to satisfy different psychological needs. These needs are typically categorised as instrumental (task-focused), hedonic (pleasure-oriented), social (connection-based), and emotional (coping-related) (Sundar &amp; Limperos, 2013). These motivations explain how phones are integrated into daily life and how they influence wellbeing (Davis, 1989).

== Instrumental and pragmatic motivations ==
{{RoundBoxTop|theme=1}}[[File:Boy from vector.svg|30px|link=]] '''Case example: Alex'''

Alex starts each morning by checking the weather app, navigating traffic with GPS, reviewing lecture notes, and setting assignments reminders. Each action is purposeful and task-oriented. His smartphone works as a practical assistant, helping him manage academic and daily tasks.
{{RoundBoxBottom}}

Instrumental motivations describe using phones to achieve specific goals. These motivations support productivity, scheduling, navigation, and information access (Joo &amp; Sang, 2013). Pragmatic motivations are a subtype of instrumental motivations, focusing on immediate benefits such as reassurance during emergencies or quick information access (Meng et al., 2020). Together, these motivations position smartphones as essential tools for managing everyday life (Wilson et al., 2022).

[[File:Online calendar used for scheduling.png|thumb|''Figure 2.'' An example of an online calendar used daily to organise and schedule tasks.|300x300px]]'''Theoretical explanation'''

Two psychological models explain these motivations.
*The [[wikipedia:Technology_acceptance_model|technology acceptance model]] (TAM): TAM proposes that technology use depends on perceived usefulness and ease of use (Davis, 1989). When phones are both effective and effortless, they are more likely to be integrated into daily routines. This aligns with the [[wikipedia:Theory_of_reasoned_action|theory of reasoned action]], which views attitudes as strong predictors of behaviour (Fishbein &amp; Ajzen, 1975).
*The [[wikipedia:Uses_and_gratifications_theory|uses and gratifications theory]] (UGT): UGT views smartphone users as active decision-makers who adopt media to fulfil personal needs (Katz et al., 1973). Instrumental use reflects utility-based gratifications, such as finding directions, setting reminders, or contacting help (Park et al., 2009).

These models show how technology design and user choice shape instrumental motivation.

'''Research evidence'''

Instrumental phone use is consistently linked to positive outcomes. Purposeful engagement such as accessing information, improving productivity, and navigating with GPS supports daily functioning (Wilson et al., 2022). For older adults, instrumental use helps coordinate activities and supports both cognitive and social functioning (Wilson et al., 2022). Pragmatic motivations extend these benefits by providing reassurance and safety, especially during emergencies (Meng et al., 2020). From TAM's perspective, these outcomes illustrate how perceived usefulness encourages routine phone behaviour (Davis, 1989).

Meng et al. (2020) found that instrumental use is less likely to lead to dependency compared to hedonic or emotional motivations. This suggests instrumental use promotes a balanced engagement with smartphones. However, risks exist. An over-reliance on GPS may impair spatial awareness (Ruginski et al., 2019), and habitual checking of reminders can weaken memory and problem-solving skills (Elhai et al., 2017). UGT helps explain this shift from adaptive to maladaptive outcomes. While users initially select tools for convenience, {{g]} however reliance can turn into habit, diminishing autonomy and flexibility (Kardefelt-Winther, 2014).

'''Summary'''

Instrumental and pragmatic motivations generally support adaptive outcomes. Phones act as practical assistants, helping people stay organised, informed, and safe. TAM and UGT show that when phones are useful and easy to use, they integrate smoothly into daily life. Yet, the same convenience may reduce independence if it replaces critical skills. These motivations are most beneficial when phones complement rather than replace human capabilities (Joo &amp; Sang, 2013).

== Hedonic and entertainment motivations ==
{{RoundBoxTop|theme=2}}[[File:Girl silhouette.svg|20px|link=]] '''Case example: Sarah'''

After a long day of studying, Sarah relaxes by watching videos on TikTok or playing mobile games like Clash of Clans. These activities improve her mood and distract her from her current stress. But sometimes these activities delay her sleep and cut into her study time.
{{RoundBoxBottom}}

[[File:Young girl Using Her Phone.jpg|thumb|''Figure 3.'' A young woman engaged in using her mobile phone.]]

Hedonic motivations describe using phones for enjoyment, escape, and relief from boredom. This includes watching videos, gaming, scrolling [[wikipedia:Social_media|social media]], or listening to music. These activities provide quick pleasure and immersion (Granic et al., 2014).    
'''Theoretical explanation'''

Three psychological perspectives explain how hedonic motivations drive mobile phone use.
*The uses and gratifications theory: People actively select media to meet needs such as entertainment, diversion, and relaxation (Katz et al., 1973). Smartphones amplify these gratifications because they combine many entertainment functions in one portable device (Sundar &amp; Limperos, 2013).
*[[wikipedia:Self-determination_theory|Self-determination theory (SDT)]]: Hedonic use can satisfy the basic psychological needs for autonomy and competence, enhancing intrinsic motivation (Deci &amp; Ryan, 1985) For example, choosing a playlist supports autonomy. Beating a level in a game provides competence. Sharing a video with friends strengthens relatedness. When these needs are met, hedonic use can enhance wellbeing (Jeno et al., 2017).
*The compensatory internet use theory (CIUT): People use online entertainment to cope with stress, loneliness, or unmet needs (Kardefelt-Winther, 2014). For students like Sarah, this may mean escaping academic pressure through binge-watching or gaming. While this coping strategy may provide temporary relief, it can also become maladaptive if it replaces healthier habits like rest or social interaction (Elhai et al., 2019).

Together, these theories show that hedonic use is about mood regulation, need fulfilment, and distraction.

'''Research evidence'''

Short bursts of gaming or video watching can reduce stress and help recovery from daily strain (Reinecke &amp; Eden, 2016). Adolescents who engage in these activities with peers also report stronger friendships and an increased sense of belonging (Granic et al., 2014). From the perspective of self-determination theory, these hedonic activities satisfy psychological needs for competence and relatedness, supporting wellbeing when used in moderation (Deci &amp; Ryan, 2000).

However, risks arise when hedonic use becomes excessive. A meta-analysis by Kuss et al. (2018) found a moderate association between hedonic phone use and poorer sleep quality. This corroborates findings from national survey data. For instance, 70% of Australian teenagers use their phones in bed, with half of them reporting shorter sleep durations (Sleep Health Foundation, 2022). The compensatory internet use theory helps explain this pattern. While entertainment can offer short-term stress relief, excessive reliance may reinforce avoidance coping, negatively affecting wellbeing over time (Elhai et al., 2017).

The uses and gratifications theory further clarifies these risks by showing how entertainment use can shift from intentional choice to automatic checking. This diminishes autonomy and reinforces distraction habits (Sundar &amp; Limperos, 2013). Similarly, the self-determination theory suggests that hedonic use dominated by stress or obligation fails to satisfy basic needs and may undermine health (Deci &amp; Ryan, 1985).

'''Summary'''

Hedonic motivations highlights the double-edged nature of mobile entertainment. When balanced, these activities fulfil psychological needs and aid recovery from stress (Reinecke &amp; Eden, 2016). Conversely, when driven by avoidance or habit, they can impair sleep, learning, and overall wellbeing (Kuss et al., 2018). The impact depends less on the technology itself and more on whether people use it as a balanced form of enjoyment or as a substitute for healthier coping strategies.

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[[File:Thought_bubble.svg|40px|link=]]
'''Reflection box: How do you use your phone for entertainment?'''

Think about the last time you used your phone just for enjoyment (e.g., watching videos, playing games, scrolling social media). 

*Did it help you feel more relaxed or energised?
*Did it ever interfere with your sleep?
*What needs (autonomy, competence, connection) do you think were met by that phone use?
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== Social motivations ==
{{RoundBoxTop|theme=3}}
[[File:Girl silhouette.svg|20px|link=]]
'''Case example: Taylor'''

Taylor recently moved to a new city for university. She uses messaging apps, video calls, and social media to stay in touch with old friends. These interactions help her feel connected to her community, despite the distance.
{{RoundBoxBottom}}

Taylor’s behaviour reflects social motivations. These motivations stem from the human need for connection, belonging, and validation. Phones allow people to maintain relationships, share updates and feel included regardless of distance. Social motivations include messaging, posting content, and joining online groups. While often rewarding, social use can also create pressure to stay available and can increase stress when users feel excluded (Kim &amp; Lee, 2011).  

[[File:Online course, fall 2020, Senior Citizens Write Wikipedia.png|thumb|right|''Figure 4''. A group of friends using video calls to socialise with each other.|300x300px]]

'''Theoretical explanation'''

Three psychological models explain how social motivations drive mobile phone use.
*Self-determination theory: SDT identifies relatedness as a basic psychological need, alongside autonomy and competence (Deci &amp; Ryan, 2000). Phones support relatedness by offering frequent contact (Joo &amp; Sang, 2013). However, when connections are pursued out of obligation or fear of exclusion, the benefits of relatedness decline (Kushlev et al., 2016).
*Uses and gratifications theory: UGT sees users as active decision-makers who choose media that fulfils needs such as belonging, reassurance, or identity expression (Katz et al., 1973). Phones make this easy by providing instant connection.
*Compensatory Internet Use Theory (CIUT): CIUT proposes that people use online communication to compensate for offline social challenges (Kardefelt-Winther, 2014). For example, students may rely on group chats or social media to reduce loneliness or social anxiety. While this can provide temporary comfort, it may also reinforce dependence and reduce engagement in offline interactions.

Together, these perspectives suggest that social motivations can be both supportive and anxiety-driven.

'''Research evidence'''

Research highlights both benefits and risks associated with social phone use. On the positive side, social interactions via smartphones strengthen perceived support and bonding. Tools like messaging, [[wikipedia:Videotelephony|video calls]], and [[wikipedia:Chat_room|group chats]] help reduce loneliness and help students adapt to new environments (Joo &amp; Sang, 2013). Surveys show that sharing humour, updates, and encouragement online is linked with higher wellbeing and a stronger sense of belonging (Mission Australia, 2022). These benefits are strongest when social media use is intentional. Sundar and Limperos (2013) found that focusing on meaningful exchanges rather than constant scrolling reduces loneliness and supports resilience. From the perspective of SDT, this shows how social needs for relatedness can be fulfilled in autonomy-supportive ways (Deci &amp; Ryan, 2000).

However, risks arise when social use is excessive or driven by anxiety. [[wikipedia:Fear_of_missing_out|FoMo]] is strongly linked to compulsive checking, increased stress, and sleep disturbances (Przybylski et al., 2013). Longitudinal studies show that students who mainly use phones to alleviate social anxiety report heightened distress over time (Elhai et al., 2017). This aligns with the CIUT, which suggests that digital connection can serve as avoidance coping. It offers temporary relief but reinforces dependence (Kardefelt-Winther, 2014). UGT further explains that repeated reliance on phones for reassurance fosters habitual checking rather than active coping, increasing vulnerability to problematic use (Kuss et al., 2018).

'''Summary'''

Social motivations demonstrate how phones serve as tools for belonging. When use is selective and autonomy-supportive, phones enhance wellbeing by fulfilling relatedness needs (Deci &amp; Ryan, 2000). Conversely, when driven by FoMo or avoidance, social phone use may foster stress, dependence, and reduced offline interaction (Elhai et al., 2019). A positive balance depends on whether social phone use supports authentic connection or substitutes for healthier coping strategies.

== Emotional and mood regulatory motivations ==
{{RoundBoxTop|theme=4}}
[[File:Boy from vector.svg|30px|link=]]
'''Case example: Sam'''

Sam often feels anxious before giving class presentations. To cope, he checks his phone for supportive messages and scrolls through Instagram to distract himself. This brings temporary relief but prevents him from addressing the root of his anxiety.
{{RoundBoxBottom}}

Sam’s behaviour reflects emotional and mood regulatory motivations, where phones are used to manage stress, loneliness, anxiety, sadness, or boredom. Smartphones provide immediate outlets for distraction, reassurance, or self-expression. This helps stabilise mood in the short term. These motivations stem from the basic psychological drive to regulate emotions and regain a sense of control. However, reliance on phones can shift from adaptive coping to maladaptive dependence when overused.

'''Theoretical explanation'''

Three psychological models explain these dynamics.
*[[wikipedia:Emotional_self-regulation|Emotion regulation]] theory: Gross (1998) distinguishes strategies such as distraction, reappraisal, and suppression. Phones support distraction (e.g., scrolling feeds) and can encourage reappraisal by offering uplifting or humorous content.
*[[wikipedia:Mood_management_theory|Mood management theory]]: Zillmann (1988) suggests people choose media to reduce negative states or maintain positive ones. Smartphones provide instant access to entertainment, games, and social contact, making them powerful tools for mood adjustment.
*Compensatory internet use theory: Kardefelt-Winther (2014) proposes that individuals turn to media to offset offline stressors. While this can relieve tension, it may also fuel avoidance if deeper problems remain unresolved.

Together, these theories explain why emotionally motivated phone use can bring short-term relief but also carry long-term risks.

'''Research evidence'''

Moderate use of relaxation apps, mindfulness tools, or supportive messaging can reduce stress and improve wellbeing in the short term (Reinecke &amp; Eden, 2016). Social sharing and online self-expression offer [[wikipedia:Catharsis|catharsis]] and strengthen social bonds, consistent with mood management theory (Kim &amp; Lee, 2011). Additional evidence shows that reappraisal through positive online content may enhance emotional resilience (Park &amp; Valenzuela, 2009).

However, risks arise when emotional regulation relies too heavily on smartphones. Escaping stress through constant phone use has been linked to procrastination, avoidance, and increased stress (Elhai et al., 2019). Heavy nighttime use is consistently linked with poor sleep quality and heightened anxiety (Sleep Health Foundation, 2022). Longitudinal studies further show that depending on phones as a primary coping mechanism predicts worsening depressive symptoms over time, especially in adolescents (Kuss et al., 2018). These findings illustrate the shift from adaptive distraction to maladaptive dependence when stressors remain unaddressed.

[[File:Young boy struggling to sleep.png|thumb|''Figure 5.'' Illustration of a young boy struggling to fall asleep at night because he is using his phone before bed.]]'''Summary'''

Emotion regulation theory highlights how phones provide quick access to distraction and reappraisal. Mood management theory explains why people seek uplifting or entertaining content. Compensatory internet use theory clarifies why these strategies sometimes slip into avoidance and dependency. Using digital tools is most beneficial when they complement healthy offline coping strategies such as problem-solving, rest, and social support, rather than replacing them (Kushlev et al., 2016).

== Practical implications ==
To promote healthier smartphone use, strategies must address the different motivations that drive behaviour. Grounding these strategies in psychological theory ensures that they move beyond surface-level advice and instead build lasting skills for wellbeing. By tailoring strategies, smartphone use can shift from automatic or avoidant behaviours to intentional habits that enhance autonomy, competence, and relatedness (Deci &amp; Ryan, 2000).

'''Instrumental use'''

Smartphones are valuable tools for organisation, navigation, and learning. Using calendar apps, focus timers, and note-taking platforms can strengthen autonomy and competence (Deci &amp; Ryan, 2000). Practical strategies could include driving offline routes to maintain spatial awareness while reducing an over-reliance on devices like GPS (Ruginski et al., 2019). Also, reflecting on which apps genuinely support goals helps build self-regulation (Hadlington, 2015). Workplaces, schools, and families can also encourage digital literacy training to help people use apps more intentionally, aligning with self-regulation theory, which emphasises feedback and planning in achieving goals.

'''Hedonic use'''

Phones provide entertainment and while these activities can improve mood, excessive use can disrupt sleep and concentration (Exelmans &amp; Van den Bulck, 2016). Practical strategies include setting app timers, scheduling intentional leisure breaks, and prioritising stimulating media over passive media (Sundar &amp; Limperos, 2013). Choosing creativity-focused or interactive content enhances wellbeing. Also reducing device use before sleep supports circadian rhythms and reduces fatigue (Exelmans &amp; Van den Bulck, 2016). From the perspective of mood management theory (Zillmann, 1988), these strategies work by channelling hedonic needs into intentional rather than automatic behaviours. This allows people to enjoy pleasure while minimising costs to health and productivity.

'''Social use'''

Connection with others is a central reason for phone use. Structured strategies, such as curated group chats, online communities, or family check-ins, allow people to maintain relationships without constant checking. Intentional social use through prioritising direct communication over passive scrolling strengthens bonds and wellbeing (Beyens et al., 2020). Reducing notifications or batching social media checks can lower compulsive behaviours (Przybylski et al., 2013). Reflecting on whether engagement arises from genuine interest or obligation helps protect autonomy in relationships (Kushlev et al., 2016). Compensatory Internet Use Theory (Kardefelt-Winther, 2014) suggests that many turn to digital connection to offset stress or loneliness. Pairing online interactions with offline opportunities for belonging, such as community groups or shared hobbies, can reduce dependence while still supporting relatedness.

[[File:A young girl listening to music on her phone.png|thumb|''Figure 6.'' An illustration of a young girl listening to music on her mobile phone.]]

'''Emotional use'''

Smartphones are often used to cope with stress, sadness, or boredom. Adaptive strategies include guided relaxation apps, calming audio, or mood-tracking tools, which can reduce stress in the short term (Reinecke &amp; Eden, 2016). However, coping is most effective when digital tools are paired with offline support such as rest, social support, or problem-solving (Gross, 1998). Reflective practices such as asking “Is this helping me cope, or am I avoiding?” can prevent avoidance cycles linked with poor sleep and heightened anxiety (Elhai et al., 2017). Mood management theory (Zillmann, 1988) {{ic|Not in References}} explains why uplifting content helps regulate emotions. While compensatory internet use theory highlights the risks of relying on avoidance strategies without addressing underlying stressors.

==Conclusion==
Smartphone use is driven by instrumental, hedonic, social, and emotional motivations, reflecting basic psychological needs. Psychological theories like TAM, SDT, and UGT show that phones are not inherently good or bad. Their impact depends on the motivations guiding engagement. Smartphones can support productivity, enjoyment, connection, and stress relief, but the same patterns may also foster avoidance, dependency, and reduced wellbeing. This dual role highlights the importance of intentional use. Hedonic use is most rewarding when balanced, and social connection most beneficial when authentic. Instrumental and emotional use require reflection to avoid over-reliance. Recognising personal motivations helps individuals use smartphones as tools for growth, protecting autonomy, rest, and wellbeing. By understanding these dynamics, people can shape technology into an ally for resilience rather than a source of distraction.

==See also==
* [[Motivation and emotion/Book/2021/Boredom and technology addiction|Boredom and technology addiction]] (Book chapter, 2021)
* [[Motivation and emotion/Book/2016/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2016)
* [[Motivation and emotion/Book/2017/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2017) 
* [[Motivation and emotion/Book/2023/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2023)

==References==
{{Hanging indent|1=

Beyens, I., Pouwels, J. L., van Driel, I. I., Keijsers, L., &amp; Valkenburg, P. M. (2020). The effect of social media on well-being differs from adolescent to adolescent. ''Scientific Reports'', ''10'', 10763. https://doi.org/10.1038/s41598-020-67727-7

Davis, F. D. (1989). Perceived usefulness, perceived ease of use, and user acceptance of information technology. ''MIS Quarterly'', ''13''(3), 319–340. https://doi.org/10.2307/249008

Deci, E. L., &amp; Ryan, R. M. (1985). ''Intrinsic motivation and self-determination in human behavior''. Plenum Press. https://doi.org/10.1007/978-1-4899-2271-7

Deci, E. L., &amp; Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. ''Psychological Inquiry'', ''11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Elhai, J. D., Levine, J. C., Dvorak, R. D., &amp; Hall, B. J. (2017). Non-social features of smartphone use are most related to depression, anxiety and problematic smartphone use. ''Computers in Human Behavior'', ''69'', 75–82. https://doi.org/10.1016/j.chb.2016.12.023

Elhai, J. D., Yang, H., McKay, D., &amp; Asmundson, G. J. (2019). Depression and anxiety symptoms are related to problematic smartphone use severity in Chinese young adults: Fear of missing out as a mediator. ''Addictive Behaviors'', ''101'', 105962. https://doi.org/10.1016/j.addbeh.2019.04.020

Exelmans, L., &amp; Van den Bulck, J. (2016). Bedtime mobile phone use and sleep in adults. ''Social Science &amp; Medicine'', ''148'', 93–101. https://doi.org/10.1016/j.socscimed.2015.11.037

Fishbein, M., &amp; Ajzen, I. (1975). ''Belief, attitude, intention, and behavior: An introduction to theory and research''. Addison-Wesley. https://people.umass.edu/aizen/f&amp;a1975.html

Granic, I., Lobel, A., &amp; Engels, R. C. M. E. (2014). The benefits of playing video games. ''American Psychologist'', ''69''(1), 66–78. https://doi.org/10.1037/a0034857

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/10.1037/1089-2680.2.3.271

Hadlington, L. (2015). Cognitive failures in daily life: Exploring the link with Internet addiction and problematic mobile phone use. ''Computers in Human Behavior'', ''51'', 75–81. https://doi.org/10.1016/j.chb.2015.04.036

Joo, T. M., &amp; Sang, Y. (2013). Exploring Koreans’ smartphone usage: An integrated model of the technology acceptance model and uses and gratifications theory. ''Computers in Human Behavior'', ''29''(6), 2512–2518. https://doi.org/10.1016/j.chb.2013.06.002

Kardefelt-Winther, D. (2014). A conceptual and methodological critique of Internet addiction research: Towards a model of compensatory Internet use. ''Computers in Human Behavior'', ''31'', 351–354. https://doi.org/10.1016/j.chb.2013.10.059

Katz, E., Blumler, J. G., &amp; Gurevitch, M. (1973). Uses and gratifications research. ''Public Opinion Quarterly'', ''37''(4), 509–523. https://doi.org/10.1086/268109

Kim, J., &amp; Lee, J. E. R. (2011). The Facebook paths to happiness: Effects of the number of Facebook friends and self-presentation on subjective well-being. ''Cyberpsychology, Behavior, and Social Networking'', ''14''(6), 359–364. https://doi.org/10.1089/cyber.2010.0374

Kushlev, K., Proulx, J., &amp; Dunn, E. W. (2016). Digitally connected, socially disconnected: The effects of relying on technology rather than other people. ''Computers in Human Behavior'', ''58'', 140–148. https://doi.org/10.1016/j.chb.2017.07.001

Kuss, D. J., Griffiths, M. D., Karila, L., &amp; Billieux, J. (2018). Internet addiction: A systematic review of epidemiological research for the last decade. ''Current Pharmaceutical Design'', ''20''(25), 4026–4052. https://doi.org/10.2174/13816128113199990617

Meng, H., Cao, H., Hao, R., Zhou, N., Liang, Y., Wu, L., Jiang, L., Ma, R., Li, B., Deng, L., Lin, Z., Lin, X., &amp; Zhang, J. (2020). Smartphone use motivation and problematic smartphone use in a national representative sample of Chinese adolescents: The mediating roles of smartphone use time for various activities. ''Journal of behavioral addictions'', ''9''(1), 163–174. https://doi.org/10.1556/2006.2020.00004

Park, N., Kee, K. F., &amp; Valenzuela, S. (2009). Being immersed in social networking environment: Facebook groups, uses and gratifications, and social outcomes. ''Cyberpsychology, Behavior, and Social Networking'', ''13''(6), 357–360. https://doi.org/10.1089/cpb.2009.0003

Przybylski, A. K., Murayama, K., DeHaan, C. R., &amp; Gladwell, V. (2013). Motivational, emotional, and behavioral correlates of fear of missing out. ''Computers in Human Behavior'', ''29''(4), 1841–1848. https://doi.org/10.1016/j.chb.2013.02.014

Reinecke, L., &amp; Eden, A. (2016). Media use and recreation: Media-induced recovery as a link between media exposure and well-being. In L. Reinecke &amp; M. B. Oliver (Eds.), ''The Routledge handbook of media use and well-being'' (pp. 106–117). Routledge. https://doi.org/10.4324/9781315714752

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., &amp; Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. ''Journal of Environmental Psychology'', ''64'', 12–20. https://doi.org/10.1016/j.jenvp.2019.05.001

Sundar, S. S., &amp; Limperos, A. M. (2013). Uses and grats 2.0: New gratifications for new media. ''Journal of Broadcasting &amp; Electronic Media'', ''57''(4), 504–525. https://doi.org/10.1080/08838151.2013.845827

Wilson, S. A., Byrne, P., Rodgers, S. E., &amp; Maden, M. (2022). A systematic review of smartphone and tablet use by older adults with and without cognitive impairment. ''Innovation in Aging'', ''6''(2), igac002. https://doi.org/10.1093/geroni/igac002

}}

==External links==
* [https://www.sleephealthfoundation.org.au/sleep-topics/technology-and-sleep Technology and sleep] (Sleep Health Foundation)
* [https://www.missionaustralia.com.au/publications/youth-survey/state-reports-2022?layout=columns Youth survey report 2022] (Mission Australia)

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== Overview ==
{{RoundBoxTop|theme=3}}
[[File:Phone use on a train.png|thumb|''' ''Figure 1''.''' People using their phone on the train.]]

'''Imagine this ...'''

You are sitting on a crowded train after a long day. A young boy nearby laughs at a meme on Instagram. A woman beside him scrolls through her messages. Across the aisle, a student types reminders into a calendar app. Next to you, a man endlessly scrolls through social media updates. What draws each of these people to their phone? Is it comfort, connection, distraction, or control? These are everyday actions, yet behind them lie hidden motivations that shape our relationship with smartphones.
{{RoundBoxBottom}}

[[wikipedia:Mobile_phone|Mobile phones]] are central to modern life, influencing how people work, connect, and cope with [[wikipedia:Stress_(biology)|stress]]. Their use is driven by complex psychological needs that affect daily [[wikipedia:Behavior|behaviour]] and [[wikipedia:Well-being|wellbeing]]. Understanding these [[wikipedia:Motivation|motivations]] reveals why people engage with phones in unique ways, and why these behaviours can bring both benefits and challenges. [[wikipedia:Smartphone|Smartphones]] enhance productivity, entertainment, and social bonds. Yet, excessive or compulsive use can lead to [[wikipedia:Procrastination|procrastination]], [[wikipedia:Anxiety|anxiety]], and emotional [[wikipedia:Avoidance_coping|avoidance]]. This chapter explores how motivations shape phone use and offers strategies for healthier, intentional smartphone use that supports wellbeing.

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What motivates people to use smartphones in their daily lives?
* What psychological theories explain these motivations?
* How do different motivations impact wellbeing in positive and negative ways?
* What strategies can encourage mindful and beneficial phone use?
{{RoundBoxBottom}}

== Key motivations for mobile phone use ==
People use mobile phones to satisfy different psychological needs. These needs are typically categorised as instrumental (task-focused), hedonic (pleasure-oriented), social (connection-based), and emotional (coping-related) (Sundar &amp; Limperos, 2013). These motivations explain how phones are integrated into daily life and how they influence wellbeing (Davis, 1989).

== Instrumental and pragmatic motivations ==
{{RoundBoxTop|theme=1}}[[File:Boy from vector.svg|30px|link=]] '''Case example: Alex'''

Alex starts each morning by checking the weather app, navigating traffic with GPS, reviewing lecture notes, and setting assignments reminders. Each action is purposeful and task-oriented. His smartphone works as a practical assistant, helping him manage academic and daily tasks.
{{RoundBoxBottom}}

Instrumental motivations describe using phones to achieve specific goals. These motivations support productivity, scheduling, navigation, and information access (Joo &amp; Sang, 2013). Pragmatic motivations are a subtype of instrumental motivations, focusing on immediate benefits such as reassurance during emergencies or quick information access (Meng et al., 2020). Together, these motivations position smartphones as essential tools for managing everyday life (Wilson et al., 2022).

[[File:Online calendar used for scheduling.png|thumb|''Figure 2.'' An example of an online calendar used daily to organise and schedule tasks.|300x300px]]'''Theoretical explanation'''

Two psychological models explain these motivations.
*The [[wikipedia:Technology_acceptance_model|technology acceptance model]] (TAM): TAM proposes that technology use depends on perceived usefulness and ease of use (Davis, 1989). When phones are both effective and effortless, they are more likely to be integrated into daily routines. This aligns with the [[wikipedia:Theory_of_reasoned_action|theory of reasoned action]], which views attitudes as strong predictors of behaviour (Fishbein &amp; Ajzen, 1975).
*The [[wikipedia:Uses_and_gratifications_theory|uses and gratifications theory]] (UGT): UGT views smartphone users as active decision-makers who adopt media to fulfil personal needs (Katz et al., 1973). Instrumental use reflects utility-based gratifications, such as finding directions, setting reminders, or contacting help (Park et al., 2009).

These models show how technology design and user choice shape instrumental motivation.

'''Research evidence'''

Instrumental phone use is consistently linked to positive outcomes. Purposeful engagement such as accessing information, improving productivity, and navigating with GPS supports daily functioning (Wilson et al., 2022). For older adults, instrumental use helps coordinate activities and supports both cognitive and social functioning (Wilson et al., 2022). Pragmatic motivations extend these benefits by providing reassurance and safety, especially during emergencies (Meng et al., 2020). From TAM's perspective, these outcomes illustrate how perceived usefulness encourages routine phone behaviour (Davis, 1989).

Meng et al. (2020) found that instrumental use is less likely to lead to dependency compared to hedonic or emotional motivations. This suggests instrumental use promotes a balanced engagement with smartphones. However, risks exist. An over-reliance on GPS may impair spatial awareness (Ruginski et al., 2019), and habitual checking of reminders can weaken memory and problem-solving skills (Elhai et al., 2017). UGT helps explain this shift from adaptive to maladaptive outcomes. While users initially select tools for convenience, {{g]} however reliance can turn into habit, diminishing autonomy and flexibility (Kardefelt-Winther, 2014).

'''Summary'''

Instrumental and pragmatic motivations generally support adaptive outcomes. Phones act as practical assistants, helping people stay organised, informed, and safe. TAM and UGT show that when phones are useful and easy to use, they integrate smoothly into daily life. Yet, the same convenience may reduce independence if it replaces critical skills. These motivations are most beneficial when phones complement rather than replace human capabilities (Joo &amp; Sang, 2013).

== Hedonic and entertainment motivations ==
{{RoundBoxTop|theme=2}}[[File:Girl silhouette.svg|20px|link=]] '''Case example: Sarah'''

After a long day of studying, Sarah relaxes by watching videos on TikTok or playing mobile games like Clash of Clans. These activities improve her mood and distract her from her current stress. But sometimes these activities delay her sleep and cut into her study time.
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[[File:Young girl Using Her Phone.jpg|thumb|''Figure 3.'' A young woman engaged in using her mobile phone.]]

Hedonic motivations describe using phones for enjoyment, escape, and relief from boredom. This includes watching videos, gaming, scrolling [[wikipedia:Social_media|social media]], or listening to music. These activities provide quick pleasure and immersion (Granic et al., 2014).    
'''Theoretical explanation'''

Three psychological perspectives explain how hedonic motivations drive mobile phone use.
*The uses and gratifications theory: People actively select media to meet needs such as entertainment, diversion, and relaxation (Katz et al., 1973). Smartphones amplify these gratifications because they combine many entertainment functions in one portable device (Sundar &amp; Limperos, 2013).
*[[wikipedia:Self-determination_theory|Self-determination theory (SDT)]]: Hedonic use can satisfy the basic psychological needs for autonomy and competence, enhancing intrinsic motivation (Deci &amp; Ryan, 1985) For example, choosing a playlist supports autonomy. Beating a level in a game provides competence. Sharing a video with friends strengthens relatedness. When these needs are met, hedonic use can enhance wellbeing (Jeno et al., 2017).
*The compensatory internet use theory (CIUT): People use online entertainment to cope with stress, loneliness, or unmet needs (Kardefelt-Winther, 2014). For students like Sarah, this may mean escaping academic pressure through binge-watching or gaming. While this coping strategy may provide temporary relief, it can also become maladaptive if it replaces healthier habits like rest or social interaction (Elhai et al., 2019).

Together, these theories show that hedonic use is about mood regulation, need fulfilment, and distraction.

'''Research evidence'''

Short bursts of gaming or video watching can reduce stress and help recovery from daily strain (Reinecke &amp; Eden, 2016). Adolescents who engage in these activities with peers also report stronger friendships and an increased sense of belonging (Granic et al., 2014). From the perspective of self-determination theory, these hedonic activities satisfy psychological needs for competence and relatedness, supporting wellbeing when used in moderation (Deci &amp; Ryan, 2000).

However, risks arise when hedonic use becomes excessive. A meta-analysis by Kuss et al. (2018) found a moderate association between hedonic phone use and poorer sleep quality. This corroborates findings from national survey data. For instance, 70% of Australian teenagers use their phones in bed, with half of them reporting shorter sleep durations (Sleep Health Foundation, 2022). The compensatory internet use theory helps explain this pattern. While entertainment can offer short-term stress relief, excessive reliance may reinforce avoidance coping, negatively affecting wellbeing over time (Elhai et al., 2017).

The uses and gratifications theory further clarifies these risks by showing how entertainment use can shift from intentional choice to automatic checking. This diminishes autonomy and reinforces distraction habits (Sundar &amp; Limperos, 2013). Similarly, the self-determination theory suggests that hedonic use dominated by stress or obligation fails to satisfy basic needs and may undermine health (Deci &amp; Ryan, 1985).

'''Summary'''

Hedonic motivations highlights the double-edged nature of mobile entertainment. When balanced, these activities fulfil psychological needs and aid recovery from stress (Reinecke &amp; Eden, 2016). Conversely, when driven by avoidance or habit, they can impair sleep, learning, and overall wellbeing (Kuss et al., 2018). The impact depends less on the technology itself and more on whether people use it as a balanced form of enjoyment or as a substitute for healthier coping strategies.

{{RoundBoxTop|theme=9}}
[[File:Thought_bubble.svg|40px|link=]]
'''Reflection box: How do you use your phone for entertainment?'''

Think about the last time you used your phone just for enjoyment (e.g., watching videos, playing games, scrolling social media). 

*Did it help you feel more relaxed or energised?
*Did it ever interfere with your sleep?
*What needs (autonomy, competence, connection) do you think were met by that phone use?
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== Social motivations ==
{{RoundBoxTop|theme=3}}
[[File:Girl silhouette.svg|20px|link=]]
'''Case example: Taylor'''

Taylor recently moved to a new city for university. She uses messaging apps, video calls, and social media to stay in touch with old friends. These interactions help her feel connected to her community, despite the distance.
{{RoundBoxBottom}}

Taylor’s behaviour reflects social motivations. These motivations stem from the human need for connection, belonging, and validation. Phones allow people to maintain relationships, share updates and feel included regardless of distance. Social motivations include messaging, posting content, and joining online groups. While often rewarding, social use can also create pressure to stay available and can increase stress when users feel excluded (Kim &amp; Lee, 2011).  

[[File:Online course, fall 2020, Senior Citizens Write Wikipedia.png|thumb|right|''Figure 4''. A group of friends using video calls to socialise with each other.|300x300px]]

'''Theoretical explanation'''

Three psychological models explain how social motivations drive mobile phone use.
*Self-determination theory: SDT identifies relatedness as a basic psychological need, alongside autonomy and competence (Deci &amp; Ryan, 2000). Phones support relatedness by offering frequent contact (Joo &amp; Sang, 2013). However, when connections are pursued out of obligation or fear of exclusion, the benefits of relatedness decline (Kushlev et al., 2016).
*Uses and gratifications theory: UGT sees users as active decision-makers who choose media that fulfils needs such as belonging, reassurance, or identity expression (Katz et al., 1973). Phones make this easy by providing instant connection.
*Compensatory Internet Use Theory (CIUT): CIUT proposes that people use online communication to compensate for offline social challenges (Kardefelt-Winther, 2014). For example, students may rely on group chats or social media to reduce loneliness or social anxiety. While this can provide temporary comfort, it may also reinforce dependence and reduce engagement in offline interactions.

Together, these perspectives suggest that social motivations can be both supportive and anxiety-driven.

'''Research evidence'''

Research highlights both benefits and risks associated with social phone use. On the positive side, social interactions via smartphones strengthen perceived support and bonding. Tools like messaging, [[wikipedia:Videotelephony|video calls]], and [[wikipedia:Chat_room|group chats]] help reduce loneliness and help students adapt to new environments (Joo &amp; Sang, 2013). Surveys show that sharing humour, updates, and encouragement online is linked with higher wellbeing and a stronger sense of belonging (Mission Australia, 2022). These benefits are strongest when social media use is intentional. Sundar and Limperos (2013) found that focusing on meaningful exchanges rather than constant scrolling reduces loneliness and supports resilience. From the perspective of SDT, this shows how social needs for relatedness can be fulfilled in autonomy-supportive ways (Deci &amp; Ryan, 2000).

However, risks arise when social use is excessive or driven by anxiety. [[wikipedia:Fear_of_missing_out|FoMo]] is strongly linked to compulsive checking, increased stress, and sleep disturbances (Przybylski et al., 2013). Longitudinal studies show that students who mainly use phones to alleviate social anxiety report heightened distress over time (Elhai et al., 2017). This aligns with the CIUT, which suggests that digital connection can serve as avoidance coping. It offers temporary relief but reinforces dependence (Kardefelt-Winther, 2014). UGT further explains that repeated reliance on phones for reassurance fosters habitual checking rather than active coping, increasing vulnerability to problematic use (Kuss et al., 2018).

'''Summary'''

Social motivations demonstrate how phones serve as tools for belonging. When use is selective and autonomy-supportive, phones enhance wellbeing by fulfilling relatedness needs (Deci &amp; Ryan, 2000). Conversely, when driven by FoMo or avoidance, social phone use may foster stress, dependence, and reduced offline interaction (Elhai et al., 2019). A positive balance depends on whether social phone use supports authentic connection or substitutes for healthier coping strategies.

== Emotional and mood regulatory motivations ==
{{RoundBoxTop|theme=4}}
[[File:Boy from vector.svg|30px|link=]]
'''Case example: Sam'''

Sam often feels anxious before giving class presentations. To cope, he checks his phone for supportive messages and scrolls through Instagram to distract himself. This brings temporary relief but prevents him from addressing the root of his anxiety.
{{RoundBoxBottom}}

Sam’s behaviour reflects emotional and mood regulatory motivations, where phones are used to manage stress, loneliness, anxiety, sadness, or boredom. Smartphones provide immediate outlets for distraction, reassurance, or self-expression. This helps stabilise mood in the short term. These motivations stem from the basic psychological drive to regulate emotions and regain a sense of control. However, reliance on phones can shift from adaptive coping to maladaptive dependence when overused.

'''Theoretical explanation'''

Three psychological models explain these dynamics.
*[[wikipedia:Emotional_self-regulation|Emotion regulation]] theory: Gross (1998) distinguishes strategies such as distraction, reappraisal, and suppression. Phones support distraction (e.g., scrolling feeds) and can encourage reappraisal by offering uplifting or humorous content.
*[[wikipedia:Mood_management_theory|Mood management theory]]: Zillmann (1988) suggests people choose media to reduce negative states or maintain positive ones. Smartphones provide instant access to entertainment, games, and social contact, making them powerful tools for mood adjustment.
*Compensatory internet use theory: Kardefelt-Winther (2014) proposes that individuals turn to media to offset offline stressors. While this can relieve tension, it may also fuel avoidance if deeper problems remain unresolved.

Together, these theories explain why emotionally motivated phone use can bring short-term relief but also carry long-term risks.

'''Research evidence'''

Moderate use of relaxation apps, mindfulness tools, or supportive messaging can reduce stress and improve wellbeing in the short term (Reinecke &amp; Eden, 2016). Social sharing and online self-expression offer [[wikipedia:Catharsis|catharsis]] and strengthen social bonds, consistent with mood management theory (Kim &amp; Lee, 2011). Additional evidence shows that reappraisal through positive online content may enhance emotional resilience (Park &amp; Valenzuela, 2009).

However, risks arise when emotional regulation relies too heavily on smartphones. Escaping stress through constant phone use has been linked to procrastination, avoidance, and increased stress (Elhai et al., 2019). Heavy nighttime use is consistently linked with poor sleep quality and heightened anxiety (Sleep Health Foundation, 2022). Longitudinal studies further show that depending on phones as a primary coping mechanism predicts worsening depressive symptoms over time, especially in adolescents (Kuss et al., 2018). These findings illustrate the shift from adaptive distraction to maladaptive dependence when stressors remain unaddressed.

[[File:Young boy struggling to sleep.png|thumb|''Figure 5.'' Illustration of a young boy struggling to fall asleep at night because he is using his phone before bed.]]'''Summary'''

Emotion regulation theory highlights how phones provide quick access to distraction and reappraisal. Mood management theory explains why people seek uplifting or entertaining content. Compensatory internet use theory clarifies why these strategies sometimes slip into avoidance and dependency. Using digital tools is most beneficial when they complement healthy offline coping strategies such as problem-solving, rest, and social support, rather than replacing them (Kushlev et al., 2016).

== Practical implications ==
To promote healthier smartphone use, strategies must address the different motivations that drive behaviour. Grounding these strategies in psychological theory ensures that they move beyond surface-level advice and instead build lasting skills for wellbeing. By tailoring strategies, smartphone use can shift from automatic or avoidant behaviours to intentional habits that enhance autonomy, competence, and relatedness (Deci &amp; Ryan, 2000).

'''Instrumental use'''

Smartphones are valuable tools for organisation, navigation, and learning. Using calendar apps, focus timers, and note-taking platforms can strengthen autonomy and competence (Deci &amp; Ryan, 2000). Practical strategies could include driving offline routes to maintain spatial awareness while reducing an over-reliance on devices like GPS (Ruginski et al., 2019). Also, reflecting on which apps genuinely support goals helps build self-regulation (Hadlington, 2015). Workplaces, schools, and families can also encourage digital literacy training to help people use apps more intentionally, aligning with self-regulation theory, which emphasises feedback and planning in achieving goals.

'''Hedonic use'''

Phones provide entertainment and while these activities can improve mood, excessive use can disrupt sleep and concentration (Exelmans &amp; Van den Bulck, 2016). Practical strategies include setting app timers, scheduling intentional leisure breaks, and prioritising stimulating media over passive media (Sundar &amp; Limperos, 2013). Choosing creativity-focused or interactive content enhances wellbeing. Also reducing device use before sleep supports circadian rhythms and reduces fatigue (Exelmans &amp; Van den Bulck, 2016). From the perspective of mood management theory (Zillmann, 1988), these strategies work by channelling hedonic needs into intentional rather than automatic behaviours. This allows people to enjoy pleasure while minimising costs to health and productivity.

'''Social use'''

Connection with others is a central reason for phone use. Structured strategies, such as curated group chats, online communities, or family check-ins, allow people to maintain relationships without constant checking. Intentional social use through prioritising direct communication over passive scrolling strengthens bonds and wellbeing (Beyens et al., 2020). Reducing notifications or batching social media checks can lower compulsive behaviours (Przybylski et al., 2013). Reflecting on whether engagement arises from genuine interest or obligation helps protect autonomy in relationships (Kushlev et al., 2016). Compensatory Internet Use Theory (Kardefelt-Winther, 2014) suggests that many turn to digital connection to offset stress or loneliness. Pairing online interactions with offline opportunities for belonging, such as community groups or shared hobbies, can reduce dependence while still supporting relatedness.

[[File:A young girl listening to music on her phone.png|thumb|''Figure 6.'' An illustration of a young girl listening to music on her mobile phone.]]

'''Emotional use'''

Smartphones are often used to cope with stress, sadness, or boredom. Adaptive strategies include guided relaxation apps, calming audio, or mood-tracking tools, which can reduce stress in the short term (Reinecke &amp; Eden, 2016). However, coping is most effective when digital tools are paired with offline support such as rest, social support, or problem-solving (Gross, 1998). Reflective practices such as asking “Is this helping me cope, or am I avoiding?” can prevent avoidance cycles linked with poor sleep and heightened anxiety (Elhai et al., 2017). Mood management theory (Zillmann, 1988) {{ic|Not in References}} explains why uplifting content helps regulate emotions. While compensatory internet use theory highlights the risks of relying on avoidance strategies without addressing underlying stressors.

==Conclusion==
Smartphone use is driven by instrumental, hedonic, social, and emotional motivations, reflecting basic psychological needs. Psychological theories like TAM, SDT, and UGT show that phones are not inherently good or bad. Their impact depends on the motivations guiding engagement. Smartphones can support productivity, enjoyment, connection, and stress relief, but the same patterns may also foster avoidance, dependency, and reduced wellbeing. This dual role highlights the importance of intentional use. Hedonic use is most rewarding when balanced, and social connection most beneficial when authentic. Instrumental and emotional use require reflection to avoid over-reliance. Recognising personal motivations helps individuals use smartphones as tools for growth, protecting autonomy, rest, and wellbeing. By understanding these dynamics, people can shape technology into an ally for resilience rather than a source of distraction.

==See also==
* [[Motivation and emotion/Book/2021/Boredom and technology addiction|Boredom and technology addiction]] (Book chapter, 2021)
* [[Motivation and emotion/Book/2016/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2016)
* [[Motivation and emotion/Book/2017/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2017) 
* [[Motivation and emotion/Book/2023/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2023)

==References==
{{Hanging indent|1=

Beyens, I., Pouwels, J. L., van Driel, I. I., Keijsers, L., &amp; Valkenburg, P. M. (2020). The effect of social media on well-being differs from adolescent to adolescent. ''Scientific Reports'', ''10'', 10763. https://doi.org/10.1038/s41598-020-67727-7

Davis, F. D. (1989). Perceived usefulness, perceived ease of use, and user acceptance of information technology. ''MIS Quarterly'', ''13''(3), 319–340. https://doi.org/10.2307/249008

Deci, E. L., &amp; Ryan, R. M. (1985). ''Intrinsic motivation and self-determination in human behavior''. Plenum Press. https://doi.org/10.1007/978-1-4899-2271-7

Deci, E. L., &amp; Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. ''Psychological Inquiry'', ''11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Elhai, J. D., Levine, J. C., Dvorak, R. D., &amp; Hall, B. J. (2017). Non-social features of smartphone use are most related to depression, anxiety and problematic smartphone use. ''Computers in Human Behavior'', ''69'', 75–82. https://doi.org/10.1016/j.chb.2016.12.023

Elhai, J. D., Yang, H., McKay, D., &amp; Asmundson, G. J. (2019). Depression and anxiety symptoms are related to problematic smartphone use severity in Chinese young adults: Fear of missing out as a mediator. ''Addictive Behaviors'', ''101'', 105962. https://doi.org/10.1016/j.addbeh.2019.04.020

Exelmans, L., &amp; Van den Bulck, J. (2016). Bedtime mobile phone use and sleep in adults. ''Social Science &amp; Medicine'', ''148'', 93–101. https://doi.org/10.1016/j.socscimed.2015.11.037

Fishbein, M., &amp; Ajzen, I. (1975). ''Belief, attitude, intention, and behavior: An introduction to theory and research''. Addison-Wesley. https://people.umass.edu/aizen/f&amp;a1975.html

Granic, I., Lobel, A., &amp; Engels, R. C. M. E. (2014). The benefits of playing video games. ''American Psychologist'', ''69''(1), 66–78. https://doi.org/10.1037/a0034857

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/10.1037/1089-2680.2.3.271

Hadlington, L. (2015). Cognitive failures in daily life: Exploring the link with Internet addiction and problematic mobile phone use. ''Computers in Human Behavior'', ''51'', 75–81. https://doi.org/10.1016/j.chb.2015.04.036

Joo, T. M., &amp; Sang, Y. (2013). Exploring Koreans’ smartphone usage: An integrated model of the technology acceptance model and uses and gratifications theory. ''Computers in Human Behavior'', ''29''(6), 2512–2518. https://doi.org/10.1016/j.chb.2013.06.002

Kardefelt-Winther, D. (2014). A conceptual and methodological critique of Internet addiction research: Towards a model of compensatory Internet use. ''Computers in Human Behavior'', ''31'', 351–354. https://doi.org/10.1016/j.chb.2013.10.059

Katz, E., Blumler, J. G., &amp; Gurevitch, M. (1973). Uses and gratifications research. ''Public Opinion Quarterly'', ''37''(4), 509–523. https://doi.org/10.1086/268109

Kim, J., &amp; Lee, J. E. R. (2011). The Facebook paths to happiness: Effects of the number of Facebook friends and self-presentation on subjective well-being. ''Cyberpsychology, Behavior, and Social Networking'', ''14''(6), 359–364. https://doi.org/10.1089/cyber.2010.0374

Kushlev, K., Proulx, J., &amp; Dunn, E. W. (2016). Digitally connected, socially disconnected: The effects of relying on technology rather than other people. ''Computers in Human Behavior'', ''58'', 140–148. https://doi.org/10.1016/j.chb.2017.07.001

Kuss, D. J., Griffiths, M. D., Karila, L., &amp; Billieux, J. (2018). Internet addiction: A systematic review of epidemiological research for the last decade. ''Current Pharmaceutical Design'', ''20''(25), 4026–4052. https://doi.org/10.2174/13816128113199990617

Meng, H., Cao, H., Hao, R., Zhou, N., Liang, Y., Wu, L., Jiang, L., Ma, R., Li, B., Deng, L., Lin, Z., Lin, X., &amp; Zhang, J. (2020). Smartphone use motivation and problematic smartphone use in a national representative sample of Chinese adolescents: The mediating roles of smartphone use time for various activities. ''Journal of behavioral addictions'', ''9''(1), 163–174. https://doi.org/10.1556/2006.2020.00004

Park, N., Kee, K. F., &amp; Valenzuela, S. (2009). Being immersed in social networking environment: Facebook groups, uses and gratifications, and social outcomes. ''Cyberpsychology, Behavior, and Social Networking'', ''13''(6), 357–360. https://doi.org/10.1089/cpb.2009.0003

Przybylski, A. K., Murayama, K., DeHaan, C. R., &amp; Gladwell, V. (2013). Motivational, emotional, and behavioral correlates of fear of missing out. ''Computers in Human Behavior'', ''29''(4), 1841–1848. https://doi.org/10.1016/j.chb.2013.02.014

Reinecke, L., &amp; Eden, A. (2016). Media use and recreation: Media-induced recovery as a link between media exposure and well-being. In L. Reinecke &amp; M. B. Oliver (Eds.), ''The Routledge handbook of media use and well-being'' (pp. 106–117). Routledge. https://doi.org/10.4324/9781315714752

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., &amp; Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. ''Journal of Environmental Psychology'', ''64'', 12–20. https://doi.org/10.1016/j.jenvp.2019.05.001

Sundar, S. S., &amp; Limperos, A. M. (2013). Uses and grats 2.0: New gratifications for new media. ''Journal of Broadcasting &amp; Electronic Media'', ''57''(4), 504–525. https://doi.org/10.1080/08838151.2013.845827

Wilson, S. A., Byrne, P., Rodgers, S. E., &amp; Maden, M. (2022). A systematic review of smartphone and tablet use by older adults with and without cognitive impairment. ''Innovation in Aging'', ''6''(2), igac002. https://doi.org/10.1093/geroni/igac002

}}

==External links==
* [https://www.sleephealthfoundation.org.au/sleep-topics/technology-and-sleep Technology and sleep] (Sleep Health Foundation)
* [https://www.missionaustralia.com.au/publications/youth-survey/state-reports-2022?layout=columns Youth survey report 2022] (Mission Australia)

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[[Category:Motivation and emotion/Book/Mobile phone]]
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__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}
[[File:M100 peruskorjattu sisältä.jpg|thumb|''' ''Figure 1''.''' People using their phone on the train.]]

'''Imagine this ...'''

You are sitting on a crowded train after a long day. A young boy nearby laughs at a meme on Instagram. A woman beside him scrolls through her messages. Across the aisle, a student types reminders into a calendar app. Next to you, a man endlessly scrolls through social media updates. What draws each of these people to their phone? Is it comfort, connection, distraction, or control? These are everyday actions, yet behind them lie hidden motivations that shape our relationship with smartphones.
{{RoundBoxBottom}}

[[wikipedia:Mobile_phone|Mobile phones]] are central to modern life, influencing how people work, connect, and cope with [[wikipedia:Stress_(biology)|stress]]. Their use is driven by complex psychological needs that affect daily [[wikipedia:Behavior|behaviour]] and [[wikipedia:Well-being|wellbeing]]. Understanding these [[wikipedia:Motivation|motivations]] reveals why people engage with phones in unique ways, and why these behaviours can bring both benefits and challenges. [[wikipedia:Smartphone|Smartphones]] enhance productivity, entertainment, and social bonds. Yet, excessive or compulsive use can lead to [[wikipedia:Procrastination|procrastination]], [[wikipedia:Anxiety|anxiety]], and emotional [[wikipedia:Avoidance_coping|avoidance]]. This chapter explores how motivations shape phone use and offers strategies for healthier, intentional smartphone use that supports wellbeing.

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What motivates people to use smartphones in their daily lives?
* What psychological theories explain these motivations?
* How do different motivations impact wellbeing in positive and negative ways?
* What strategies can encourage mindful and beneficial phone use?
{{RoundBoxBottom}}

== Key motivations for mobile phone use ==
People use mobile phones to satisfy different psychological needs. These needs are typically categorised as instrumental (task-focused), hedonic (pleasure-oriented), social (connection-based), and emotional (coping-related) (Sundar &amp; Limperos, 2013). These motivations explain how phones are integrated into daily life and how they influence wellbeing (Davis, 1989).

== Instrumental and pragmatic motivations ==
{{RoundBoxTop|theme=1}}[[File:Boy from vector.svg|30px|link=]] '''Case example: Alex'''

Alex starts each morning by checking the weather app, navigating traffic with GPS, reviewing lecture notes, and setting assignments reminders. Each action is purposeful and task-oriented. His smartphone works as a practical assistant, helping him manage academic and daily tasks.
{{RoundBoxBottom}}

Instrumental motivations describe using phones to achieve specific goals. These motivations support productivity, scheduling, navigation, and information access (Joo &amp; Sang, 2013). Pragmatic motivations are a subtype of instrumental motivations, focusing on immediate benefits such as reassurance during emergencies or quick information access (Meng et al., 2020). Together, these motivations position smartphones as essential tools for managing everyday life (Wilson et al., 2022).

[[File:Online calendar used for scheduling.png|thumb|''Figure 2.'' An example of an online calendar used daily to organise and schedule tasks.|300x300px]]'''Theoretical explanation'''

Two psychological models explain these motivations.
*The [[wikipedia:Technology_acceptance_model|technology acceptance model]] (TAM): TAM proposes that technology use depends on perceived usefulness and ease of use (Davis, 1989). When phones are both effective and effortless, they are more likely to be integrated into daily routines. This aligns with the [[wikipedia:Theory_of_reasoned_action|theory of reasoned action]], which views attitudes as strong predictors of behaviour (Fishbein &amp; Ajzen, 1975).
*The [[wikipedia:Uses_and_gratifications_theory|uses and gratifications theory]] (UGT): UGT views smartphone users as active decision-makers who adopt media to fulfil personal needs (Katz et al., 1973). Instrumental use reflects utility-based gratifications, such as finding directions, setting reminders, or contacting help (Park et al., 2009).

These models show how technology design and user choice shape instrumental motivation.

'''Research evidence'''

Instrumental phone use is consistently linked to positive outcomes. Purposeful engagement such as accessing information, improving productivity, and navigating with GPS supports daily functioning (Wilson et al., 2022). For older adults, instrumental use helps coordinate activities and supports both cognitive and social functioning (Wilson et al., 2022). Pragmatic motivations extend these benefits by providing reassurance and safety, especially during emergencies (Meng et al., 2020). From TAM's perspective, these outcomes illustrate how perceived usefulness encourages routine phone behaviour (Davis, 1989).

Meng et al. (2020) found that instrumental use is less likely to lead to dependency compared to hedonic or emotional motivations. This suggests instrumental use promotes a balanced engagement with smartphones. However, risks exist. An over-reliance on GPS may impair spatial awareness (Ruginski et al., 2019), and habitual checking of reminders can weaken memory and problem-solving skills (Elhai et al., 2017). UGT helps explain this shift from adaptive to maladaptive outcomes. While users initially select tools for convenience, {{g]} however reliance can turn into habit, diminishing autonomy and flexibility (Kardefelt-Winther, 2014).

'''Summary'''

Instrumental and pragmatic motivations generally support adaptive outcomes. Phones act as practical assistants, helping people stay organised, informed, and safe. TAM and UGT show that when phones are useful and easy to use, they integrate smoothly into daily life. Yet, the same convenience may reduce independence if it replaces critical skills. These motivations are most beneficial when phones complement rather than replace human capabilities (Joo &amp; Sang, 2013).

== Hedonic and entertainment motivations ==
{{RoundBoxTop|theme=2}}[[File:Girl silhouette.svg|20px|link=]] '''Case example: Sarah'''

After a long day of studying, Sarah relaxes by watching videos on TikTok or playing mobile games like Clash of Clans. These activities improve her mood and distract her from her current stress. But sometimes these activities delay her sleep and cut into her study time.
{{RoundBoxBottom}}

[[File:Young girl Using Her Phone.jpg|thumb|''Figure 3.'' A young woman engaged in using her mobile phone.]]

Hedonic motivations describe using phones for enjoyment, escape, and relief from boredom. This includes watching videos, gaming, scrolling [[wikipedia:Social_media|social media]], or listening to music. These activities provide quick pleasure and immersion (Granic et al., 2014).    
'''Theoretical explanation'''

Three psychological perspectives explain how hedonic motivations drive mobile phone use.
*The uses and gratifications theory: People actively select media to meet needs such as entertainment, diversion, and relaxation (Katz et al., 1973). Smartphones amplify these gratifications because they combine many entertainment functions in one portable device (Sundar &amp; Limperos, 2013).
*[[wikipedia:Self-determination_theory|Self-determination theory (SDT)]]: Hedonic use can satisfy the basic psychological needs for autonomy and competence, enhancing intrinsic motivation (Deci &amp; Ryan, 1985) For example, choosing a playlist supports autonomy. Beating a level in a game provides competence. Sharing a video with friends strengthens relatedness. When these needs are met, hedonic use can enhance wellbeing (Jeno et al., 2017).
*The compensatory internet use theory (CIUT): People use online entertainment to cope with stress, loneliness, or unmet needs (Kardefelt-Winther, 2014). For students like Sarah, this may mean escaping academic pressure through binge-watching or gaming. While this coping strategy may provide temporary relief, it can also become maladaptive if it replaces healthier habits like rest or social interaction (Elhai et al., 2019).

Together, these theories show that hedonic use is about mood regulation, need fulfilment, and distraction.

'''Research evidence'''

Short bursts of gaming or video watching can reduce stress and help recovery from daily strain (Reinecke &amp; Eden, 2016). Adolescents who engage in these activities with peers also report stronger friendships and an increased sense of belonging (Granic et al., 2014). From the perspective of self-determination theory, these hedonic activities satisfy psychological needs for competence and relatedness, supporting wellbeing when used in moderation (Deci &amp; Ryan, 2000).

However, risks arise when hedonic use becomes excessive. A meta-analysis by Kuss et al. (2018) found a moderate association between hedonic phone use and poorer sleep quality. This corroborates findings from national survey data. For instance, 70% of Australian teenagers use their phones in bed, with half of them reporting shorter sleep durations (Sleep Health Foundation, 2022). The compensatory internet use theory helps explain this pattern. While entertainment can offer short-term stress relief, excessive reliance may reinforce avoidance coping, negatively affecting wellbeing over time (Elhai et al., 2017).

The uses and gratifications theory further clarifies these risks by showing how entertainment use can shift from intentional choice to automatic checking. This diminishes autonomy and reinforces distraction habits (Sundar &amp; Limperos, 2013). Similarly, the self-determination theory suggests that hedonic use dominated by stress or obligation fails to satisfy basic needs and may undermine health (Deci &amp; Ryan, 1985).

'''Summary'''

Hedonic motivations highlights the double-edged nature of mobile entertainment. When balanced, these activities fulfil psychological needs and aid recovery from stress (Reinecke &amp; Eden, 2016). Conversely, when driven by avoidance or habit, they can impair sleep, learning, and overall wellbeing (Kuss et al., 2018). The impact depends less on the technology itself and more on whether people use it as a balanced form of enjoyment or as a substitute for healthier coping strategies.

{{RoundBoxTop|theme=9}}
[[File:Thought_bubble.svg|40px|link=]]
'''Reflection box: How do you use your phone for entertainment?'''

Think about the last time you used your phone just for enjoyment (e.g., watching videos, playing games, scrolling social media). 

*Did it help you feel more relaxed or energised?
*Did it ever interfere with your sleep?
*What needs (autonomy, competence, connection) do you think were met by that phone use?
{{RoundBoxBottom}}

== Social motivations ==
{{RoundBoxTop|theme=3}}
[[File:Girl silhouette.svg|20px|link=]]
'''Case example: Taylor'''

Taylor recently moved to a new city for university. She uses messaging apps, video calls, and social media to stay in touch with old friends. These interactions help her feel connected to her community, despite the distance.
{{RoundBoxBottom}}

Taylor’s behaviour reflects social motivations. These motivations stem from the human need for connection, belonging, and validation. Phones allow people to maintain relationships, share updates and feel included regardless of distance. Social motivations include messaging, posting content, and joining online groups. While often rewarding, social use can also create pressure to stay available and can increase stress when users feel excluded (Kim &amp; Lee, 2011).  

[[File:Online course, fall 2020, Senior Citizens Write Wikipedia.png|thumb|right|''Figure 4''. A group of friends using video calls to socialise with each other.|300x300px]]

'''Theoretical explanation'''

Three psychological models explain how social motivations drive mobile phone use.
*Self-determination theory: SDT identifies relatedness as a basic psychological need, alongside autonomy and competence (Deci &amp; Ryan, 2000). Phones support relatedness by offering frequent contact (Joo &amp; Sang, 2013). However, when connections are pursued out of obligation or fear of exclusion, the benefits of relatedness decline (Kushlev et al., 2016).
*Uses and gratifications theory: UGT sees users as active decision-makers who choose media that fulfils needs such as belonging, reassurance, or identity expression (Katz et al., 1973). Phones make this easy by providing instant connection.
*Compensatory Internet Use Theory (CIUT): CIUT proposes that people use online communication to compensate for offline social challenges (Kardefelt-Winther, 2014). For example, students may rely on group chats or social media to reduce loneliness or social anxiety. While this can provide temporary comfort, it may also reinforce dependence and reduce engagement in offline interactions.

Together, these perspectives suggest that social motivations can be both supportive and anxiety-driven.

'''Research evidence'''

Research highlights both benefits and risks associated with social phone use. On the positive side, social interactions via smartphones strengthen perceived support and bonding. Tools like messaging, [[wikipedia:Videotelephony|video calls]], and [[wikipedia:Chat_room|group chats]] help reduce loneliness and help students adapt to new environments (Joo &amp; Sang, 2013). Surveys show that sharing humour, updates, and encouragement online is linked with higher wellbeing and a stronger sense of belonging (Mission Australia, 2022). These benefits are strongest when social media use is intentional. Sundar and Limperos (2013) found that focusing on meaningful exchanges rather than constant scrolling reduces loneliness and supports resilience. From the perspective of SDT, this shows how social needs for relatedness can be fulfilled in autonomy-supportive ways (Deci &amp; Ryan, 2000).

However, risks arise when social use is excessive or driven by anxiety. [[wikipedia:Fear_of_missing_out|FoMo]] is strongly linked to compulsive checking, increased stress, and sleep disturbances (Przybylski et al., 2013). Longitudinal studies show that students who mainly use phones to alleviate social anxiety report heightened distress over time (Elhai et al., 2017). This aligns with the CIUT, which suggests that digital connection can serve as avoidance coping. It offers temporary relief but reinforces dependence (Kardefelt-Winther, 2014). UGT further explains that repeated reliance on phones for reassurance fosters habitual checking rather than active coping, increasing vulnerability to problematic use (Kuss et al., 2018).

'''Summary'''

Social motivations demonstrate how phones serve as tools for belonging. When use is selective and autonomy-supportive, phones enhance wellbeing by fulfilling relatedness needs (Deci &amp; Ryan, 2000). Conversely, when driven by FoMo or avoidance, social phone use may foster stress, dependence, and reduced offline interaction (Elhai et al., 2019). A positive balance depends on whether social phone use supports authentic connection or substitutes for healthier coping strategies.

== Emotional and mood regulatory motivations ==
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[[File:Boy from vector.svg|30px|link=]]
'''Case example: Sam'''

Sam often feels anxious before giving class presentations. To cope, he checks his phone for supportive messages and scrolls through Instagram to distract himself. This brings temporary relief but prevents him from addressing the root of his anxiety.
{{RoundBoxBottom}}

Sam’s behaviour reflects emotional and mood regulatory motivations, where phones are used to manage stress, loneliness, anxiety, sadness, or boredom. Smartphones provide immediate outlets for distraction, reassurance, or self-expression. This helps stabilise mood in the short term. These motivations stem from the basic psychological drive to regulate emotions and regain a sense of control. However, reliance on phones can shift from adaptive coping to maladaptive dependence when overused.

'''Theoretical explanation'''

Three psychological models explain these dynamics.
*[[wikipedia:Emotional_self-regulation|Emotion regulation]] theory: Gross (1998) distinguishes strategies such as distraction, reappraisal, and suppression. Phones support distraction (e.g., scrolling feeds) and can encourage reappraisal by offering uplifting or humorous content.
*[[wikipedia:Mood_management_theory|Mood management theory]]: Zillmann (1988) suggests people choose media to reduce negative states or maintain positive ones. Smartphones provide instant access to entertainment, games, and social contact, making them powerful tools for mood adjustment.
*Compensatory internet use theory: Kardefelt-Winther (2014) proposes that individuals turn to media to offset offline stressors. While this can relieve tension, it may also fuel avoidance if deeper problems remain unresolved.

Together, these theories explain why emotionally motivated phone use can bring short-term relief but also carry long-term risks.

'''Research evidence'''

Moderate use of relaxation apps, mindfulness tools, or supportive messaging can reduce stress and improve wellbeing in the short term (Reinecke &amp; Eden, 2016). Social sharing and online self-expression offer [[wikipedia:Catharsis|catharsis]] and strengthen social bonds, consistent with mood management theory (Kim &amp; Lee, 2011). Additional evidence shows that reappraisal through positive online content may enhance emotional resilience (Park &amp; Valenzuela, 2009).

However, risks arise when emotional regulation relies too heavily on smartphones. Escaping stress through constant phone use has been linked to procrastination, avoidance, and increased stress (Elhai et al., 2019). Heavy nighttime use is consistently linked with poor sleep quality and heightened anxiety (Sleep Health Foundation, 2022). Longitudinal studies further show that depending on phones as a primary coping mechanism predicts worsening depressive symptoms over time, especially in adolescents (Kuss et al., 2018). These findings illustrate the shift from adaptive distraction to maladaptive dependence when stressors remain unaddressed.

[[File:Young boy struggling to sleep.png|thumb|''Figure 5.'' Illustration of a young boy struggling to fall asleep at night because he is using his phone before bed.]]'''Summary'''

Emotion regulation theory highlights how phones provide quick access to distraction and reappraisal. Mood management theory explains why people seek uplifting or entertaining content. Compensatory internet use theory clarifies why these strategies sometimes slip into avoidance and dependency. Using digital tools is most beneficial when they complement healthy offline coping strategies such as problem-solving, rest, and social support, rather than replacing them (Kushlev et al., 2016).

== Practical implications ==
To promote healthier smartphone use, strategies must address the different motivations that drive behaviour. Grounding these strategies in psychological theory ensures that they move beyond surface-level advice and instead build lasting skills for wellbeing. By tailoring strategies, smartphone use can shift from automatic or avoidant behaviours to intentional habits that enhance autonomy, competence, and relatedness (Deci &amp; Ryan, 2000).

'''Instrumental use'''

Smartphones are valuable tools for organisation, navigation, and learning. Using calendar apps, focus timers, and note-taking platforms can strengthen autonomy and competence (Deci &amp; Ryan, 2000). Practical strategies could include driving offline routes to maintain spatial awareness while reducing an over-reliance on devices like GPS (Ruginski et al., 2019). Also, reflecting on which apps genuinely support goals helps build self-regulation (Hadlington, 2015). Workplaces, schools, and families can also encourage digital literacy training to help people use apps more intentionally, aligning with self-regulation theory, which emphasises feedback and planning in achieving goals.

'''Hedonic use'''

Phones provide entertainment and while these activities can improve mood, excessive use can disrupt sleep and concentration (Exelmans &amp; Van den Bulck, 2016). Practical strategies include setting app timers, scheduling intentional leisure breaks, and prioritising stimulating media over passive media (Sundar &amp; Limperos, 2013). Choosing creativity-focused or interactive content enhances wellbeing. Also reducing device use before sleep supports circadian rhythms and reduces fatigue (Exelmans &amp; Van den Bulck, 2016). From the perspective of mood management theory (Zillmann, 1988), these strategies work by channelling hedonic needs into intentional rather than automatic behaviours. This allows people to enjoy pleasure while minimising costs to health and productivity.

'''Social use'''

Connection with others is a central reason for phone use. Structured strategies, such as curated group chats, online communities, or family check-ins, allow people to maintain relationships without constant checking. Intentional social use through prioritising direct communication over passive scrolling strengthens bonds and wellbeing (Beyens et al., 2020). Reducing notifications or batching social media checks can lower compulsive behaviours (Przybylski et al., 2013). Reflecting on whether engagement arises from genuine interest or obligation helps protect autonomy in relationships (Kushlev et al., 2016). Compensatory Internet Use Theory (Kardefelt-Winther, 2014) suggests that many turn to digital connection to offset stress or loneliness. Pairing online interactions with offline opportunities for belonging, such as community groups or shared hobbies, can reduce dependence while still supporting relatedness.

[[File:A young girl listening to music on her phone.png|thumb|''Figure 6.'' An illustration of a young girl listening to music on her mobile phone.]]

'''Emotional use'''

Smartphones are often used to cope with stress, sadness, or boredom. Adaptive strategies include guided relaxation apps, calming audio, or mood-tracking tools, which can reduce stress in the short term (Reinecke &amp; Eden, 2016). However, coping is most effective when digital tools are paired with offline support such as rest, social support, or problem-solving (Gross, 1998). Reflective practices such as asking “Is this helping me cope, or am I avoiding?” can prevent avoidance cycles linked with poor sleep and heightened anxiety (Elhai et al., 2017). Mood management theory (Zillmann, 1988) {{ic|Not in References}} explains why uplifting content helps regulate emotions. While compensatory internet use theory highlights the risks of relying on avoidance strategies without addressing underlying stressors.

==Conclusion==
Smartphone use is driven by instrumental, hedonic, social, and emotional motivations, reflecting basic psychological needs. Psychological theories like TAM, SDT, and UGT show that phones are not inherently good or bad. Their impact depends on the motivations guiding engagement. Smartphones can support productivity, enjoyment, connection, and stress relief, but the same patterns may also foster avoidance, dependency, and reduced wellbeing. This dual role highlights the importance of intentional use. Hedonic use is most rewarding when balanced, and social connection most beneficial when authentic. Instrumental and emotional use require reflection to avoid over-reliance. Recognising personal motivations helps individuals use smartphones as tools for growth, protecting autonomy, rest, and wellbeing. By understanding these dynamics, people can shape technology into an ally for resilience rather than a source of distraction.

==See also==
* [[Motivation and emotion/Book/2021/Boredom and technology addiction|Boredom and technology addiction]] (Book chapter, 2021)
* [[Motivation and emotion/Book/2016/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2016)
* [[Motivation and emotion/Book/2017/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2017) 
* [[Motivation and emotion/Book/2023/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2023)

==References==
{{Hanging indent|1=

Beyens, I., Pouwels, J. L., van Driel, I. I., Keijsers, L., &amp; Valkenburg, P. M. (2020). The effect of social media on well-being differs from adolescent to adolescent. ''Scientific Reports'', ''10'', 10763. https://doi.org/10.1038/s41598-020-67727-7

Davis, F. D. (1989). Perceived usefulness, perceived ease of use, and user acceptance of information technology. ''MIS Quarterly'', ''13''(3), 319–340. https://doi.org/10.2307/249008

Deci, E. L., &amp; Ryan, R. M. (1985). ''Intrinsic motivation and self-determination in human behavior''. Plenum Press. https://doi.org/10.1007/978-1-4899-2271-7

Deci, E. L., &amp; Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. ''Psychological Inquiry'', ''11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Elhai, J. D., Levine, J. C., Dvorak, R. D., &amp; Hall, B. J. (2017). Non-social features of smartphone use are most related to depression, anxiety and problematic smartphone use. ''Computers in Human Behavior'', ''69'', 75–82. https://doi.org/10.1016/j.chb.2016.12.023

Elhai, J. D., Yang, H., McKay, D., &amp; Asmundson, G. J. (2019). Depression and anxiety symptoms are related to problematic smartphone use severity in Chinese young adults: Fear of missing out as a mediator. ''Addictive Behaviors'', ''101'', 105962. https://doi.org/10.1016/j.addbeh.2019.04.020

Exelmans, L., &amp; Van den Bulck, J. (2016). Bedtime mobile phone use and sleep in adults. ''Social Science &amp; Medicine'', ''148'', 93–101. https://doi.org/10.1016/j.socscimed.2015.11.037

Fishbein, M., &amp; Ajzen, I. (1975). ''Belief, attitude, intention, and behavior: An introduction to theory and research''. Addison-Wesley. https://people.umass.edu/aizen/f&amp;a1975.html

Granic, I., Lobel, A., &amp; Engels, R. C. M. E. (2014). The benefits of playing video games. ''American Psychologist'', ''69''(1), 66–78. https://doi.org/10.1037/a0034857

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/10.1037/1089-2680.2.3.271

Hadlington, L. (2015). Cognitive failures in daily life: Exploring the link with Internet addiction and problematic mobile phone use. ''Computers in Human Behavior'', ''51'', 75–81. https://doi.org/10.1016/j.chb.2015.04.036

Joo, T. M., &amp; Sang, Y. (2013). Exploring Koreans’ smartphone usage: An integrated model of the technology acceptance model and uses and gratifications theory. ''Computers in Human Behavior'', ''29''(6), 2512–2518. https://doi.org/10.1016/j.chb.2013.06.002

Kardefelt-Winther, D. (2014). A conceptual and methodological critique of Internet addiction research: Towards a model of compensatory Internet use. ''Computers in Human Behavior'', ''31'', 351–354. https://doi.org/10.1016/j.chb.2013.10.059

Katz, E., Blumler, J. G., &amp; Gurevitch, M. (1973). Uses and gratifications research. ''Public Opinion Quarterly'', ''37''(4), 509–523. https://doi.org/10.1086/268109

Kim, J., &amp; Lee, J. E. R. (2011). The Facebook paths to happiness: Effects of the number of Facebook friends and self-presentation on subjective well-being. ''Cyberpsychology, Behavior, and Social Networking'', ''14''(6), 359–364. https://doi.org/10.1089/cyber.2010.0374

Kushlev, K., Proulx, J., &amp; Dunn, E. W. (2016). Digitally connected, socially disconnected: The effects of relying on technology rather than other people. ''Computers in Human Behavior'', ''58'', 140–148. https://doi.org/10.1016/j.chb.2017.07.001

Kuss, D. J., Griffiths, M. D., Karila, L., &amp; Billieux, J. (2018). Internet addiction: A systematic review of epidemiological research for the last decade. ''Current Pharmaceutical Design'', ''20''(25), 4026–4052. https://doi.org/10.2174/13816128113199990617

Meng, H., Cao, H., Hao, R., Zhou, N., Liang, Y., Wu, L., Jiang, L., Ma, R., Li, B., Deng, L., Lin, Z., Lin, X., &amp; Zhang, J. (2020). Smartphone use motivation and problematic smartphone use in a national representative sample of Chinese adolescents: The mediating roles of smartphone use time for various activities. ''Journal of behavioral addictions'', ''9''(1), 163–174. https://doi.org/10.1556/2006.2020.00004

Park, N., Kee, K. F., &amp; Valenzuela, S. (2009). Being immersed in social networking environment: Facebook groups, uses and gratifications, and social outcomes. ''Cyberpsychology, Behavior, and Social Networking'', ''13''(6), 357–360. https://doi.org/10.1089/cpb.2009.0003

Przybylski, A. K., Murayama, K., DeHaan, C. R., &amp; Gladwell, V. (2013). Motivational, emotional, and behavioral correlates of fear of missing out. ''Computers in Human Behavior'', ''29''(4), 1841–1848. https://doi.org/10.1016/j.chb.2013.02.014

Reinecke, L., &amp; Eden, A. (2016). Media use and recreation: Media-induced recovery as a link between media exposure and well-being. In L. Reinecke &amp; M. B. Oliver (Eds.), ''The Routledge handbook of media use and well-being'' (pp. 106–117). Routledge. https://doi.org/10.4324/9781315714752

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., &amp; Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. ''Journal of Environmental Psychology'', ''64'', 12–20. https://doi.org/10.1016/j.jenvp.2019.05.001

Sundar, S. S., &amp; Limperos, A. M. (2013). Uses and grats 2.0: New gratifications for new media. ''Journal of Broadcasting &amp; Electronic Media'', ''57''(4), 504–525. https://doi.org/10.1080/08838151.2013.845827

Wilson, S. A., Byrne, P., Rodgers, S. E., &amp; Maden, M. (2022). A systematic review of smartphone and tablet use by older adults with and without cognitive impairment. ''Innovation in Aging'', ''6''(2), igac002. https://doi.org/10.1093/geroni/igac002

}}

==External links==
* [https://www.sleephealthfoundation.org.au/sleep-topics/technology-and-sleep Technology and sleep] (Sleep Health Foundation)
* [https://www.missionaustralia.com.au/publications/youth-survey/state-reports-2022?layout=columns Youth survey report 2022] (Mission Australia)

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[[Category:Motivation and emotion/Book/Mobile phone]]
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__TOC__

== Overview ==
{{RoundBoxTop|theme=3}}
[[File:M100 peruskorjattu sisältä.jpg|thumb|''' ''Figure 1''.''' People using their phone on the train.]]

'''Imagine this ...'''

You are sitting on a crowded train after a long day. A young boy nearby laughs at a meme on Instagram. A woman beside him scrolls through her messages. Across the aisle, a student types reminders into a calendar app. Next to you, a man endlessly scrolls through social media updates. What draws each of these people to their phone? Is it comfort, connection, distraction, or control? These are everyday actions, yet behind them lie hidden motivations that shape our relationship with smartphones.
{{RoundBoxBottom}}

[[wikipedia:Mobile_phone|Mobile phones]] are central to modern life, influencing how people work, connect, and cope with [[wikipedia:Stress_(biology)|stress]]. Their use is driven by complex psychological needs that affect daily [[wikipedia:Behavior|behaviour]] and [[wikipedia:Well-being|wellbeing]]. Understanding these [[wikipedia:Motivation|motivations]] reveals why people engage with phones in unique ways, and why these behaviours can bring both benefits and challenges. [[wikipedia:Smartphone|Smartphones]] enhance productivity, entertainment, and social bonds. Yet, excessive or compulsive use can lead to [[wikipedia:Procrastination|procrastination]], [[wikipedia:Anxiety|anxiety]], and emotional [[wikipedia:Avoidance_coping|avoidance]]. This chapter explores how motivations shape phone use and offers strategies for healthier, intentional smartphone use that supports wellbeing.

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What motivates people to use smartphones in their daily lives?
* What psychological theories explain these motivations?
* How do different motivations impact wellbeing in positive and negative ways?
* What strategies can encourage mindful and beneficial phone use?
{{RoundBoxBottom}}

== Key motivations for mobile phone use ==
People use mobile phones to satisfy different psychological needs. These needs are typically categorised as instrumental (task-focused), hedonic (pleasure-oriented), social (connection-based), and emotional (coping-related) (Sundar &amp; Limperos, 2013). These motivations explain how phones are integrated into daily life and how they influence wellbeing (Davis, 1989).

== Instrumental and pragmatic motivations ==
{{RoundBoxTop|theme=1}}[[File:Boy from vector.svg|30px|link=]] '''Case example: Alex'''

Alex starts each morning by checking the weather app, navigating traffic with GPS, reviewing lecture notes, and setting assignments reminders. Each action is purposeful and task-oriented. His smartphone works as a practical assistant, helping him manage academic and daily tasks.
{{RoundBoxBottom}}

Instrumental motivations describe using phones to achieve specific goals. These motivations support productivity, scheduling, navigation, and information access (Joo &amp; Sang, 2013). Pragmatic motivations are a subtype of instrumental motivations, focusing on immediate benefits such as reassurance during emergencies or quick information access (Meng et al., 2020). Together, these motivations position smartphones as essential tools for managing everyday life (Wilson et al., 2022).

[[File:Online calendar used for scheduling.png|thumb|''Figure 2.'' An example of an online calendar used daily to organise and schedule tasks.|300x300px]]'''Theoretical explanation'''

Two psychological models explain these motivations.
*The [[wikipedia:Technology_acceptance_model|technology acceptance model]] (TAM): TAM proposes that technology use depends on perceived usefulness and ease of use (Davis, 1989). When phones are both effective and effortless, they are more likely to be integrated into daily routines. This aligns with the [[wikipedia:Theory_of_reasoned_action|theory of reasoned action]], which views attitudes as strong predictors of behaviour (Fishbein &amp; Ajzen, 1975).
*The [[wikipedia:Uses_and_gratifications_theory|uses and gratifications theory]] (UGT): UGT views smartphone users as active decision-makers who adopt media to fulfil personal needs (Katz et al., 1973). Instrumental use reflects utility-based gratifications, such as finding directions, setting reminders, or contacting help (Park et al., 2009).

These models show how technology design and user choice shape instrumental motivation.

'''Research evidence'''

Instrumental phone use is consistently linked to positive outcomes. Purposeful engagement such as accessing information, improving productivity, and navigating with GPS supports daily functioning (Wilson et al., 2022). For older adults, instrumental use helps coordinate activities and supports both cognitive and social functioning (Wilson et al., 2022). Pragmatic motivations extend these benefits by providing reassurance and safety, especially during emergencies (Meng et al., 2020). From TAM's perspective, these outcomes illustrate how perceived usefulness encourages routine phone behaviour (Davis, 1989).

Meng et al. (2020) found that instrumental use is less likely to lead to dependency compared to hedonic or emotional motivations. This suggests instrumental use promotes a balanced engagement with smartphones. However, risks exist. An over-reliance on GPS may impair spatial awareness (Ruginski et al., 2019), and habitual checking of reminders can weaken memory and problem-solving skills (Elhai et al., 2017). UGT helps explain this shift from adaptive to maladaptive outcomes. While users initially select tools for convenience, {{g]} however reliance can turn into habit, diminishing autonomy and flexibility (Kardefelt-Winther, 2014).

'''Summary'''

Instrumental and pragmatic motivations generally support adaptive outcomes. Phones act as practical assistants, helping people stay organised, informed, and safe. TAM and UGT show that when phones are useful and easy to use, they integrate smoothly into daily life. Yet, the same convenience may reduce independence if it replaces critical skills. These motivations are most beneficial when phones complement rather than replace human capabilities (Joo &amp; Sang, 2013).

== Hedonic and entertainment motivations ==
{{RoundBoxTop|theme=2}}[[File:Girl silhouette.svg|20px|link=]] '''Case example: Sarah'''

After a long day of studying, Sarah relaxes by watching videos on TikTok or playing mobile games like Clash of Clans. These activities improve her mood and distract her from her current stress. But sometimes these activities delay her sleep and cut into her study time.
{{RoundBoxBottom}}

[[File:Young girl Using Her Phone.jpg|thumb|''Figure 3.'' A young woman engaged in using her mobile phone.]]

Hedonic motivations describe using phones for enjoyment, escape, and relief from boredom. This includes watching videos, gaming, scrolling [[wikipedia:Social_media|social media]], or listening to music. These activities provide quick pleasure and immersion (Granic et al., 2014).    
'''Theoretical explanation'''

Three psychological perspectives explain how hedonic motivations drive mobile phone use.
*The uses and gratifications theory: People actively select media to meet needs such as entertainment, diversion, and relaxation (Katz et al., 1973). Smartphones amplify these gratifications because they combine many entertainment functions in one portable device (Sundar &amp; Limperos, 2013).
*[[wikipedia:Self-determination_theory|Self-determination theory (SDT)]]: Hedonic use can satisfy the basic psychological needs for autonomy and competence, enhancing intrinsic motivation (Deci &amp; Ryan, 1985) For example, choosing a playlist supports autonomy. Beating a level in a game provides competence. Sharing a video with friends strengthens relatedness. When these needs are met, hedonic use can enhance wellbeing (Jeno et al., 2017).
*The compensatory internet use theory (CIUT): People use online entertainment to cope with stress, loneliness, or unmet needs (Kardefelt-Winther, 2014). For students like Sarah, this may mean escaping academic pressure through binge-watching or gaming. While this coping strategy may provide temporary relief, it can also become maladaptive if it replaces healthier habits like rest or social interaction (Elhai et al., 2019).

Together, these theories show that hedonic use is about mood regulation, need fulfilment, and distraction.

'''Research evidence'''

Short bursts of gaming or video watching can reduce stress and help recovery from daily strain (Reinecke &amp; Eden, 2016). Adolescents who engage in these activities with peers also report stronger friendships and an increased sense of belonging (Granic et al., 2014). From the perspective of self-determination theory, these hedonic activities satisfy psychological needs for competence and relatedness, supporting wellbeing when used in moderation (Deci &amp; Ryan, 2000).

However, risks arise when hedonic use becomes excessive. A meta-analysis by Kuss et al. (2018) found a moderate association between hedonic phone use and poorer sleep quality. This corroborates findings from national survey data. For instance, 70% of Australian teenagers use their phones in bed, with half of them reporting shorter sleep durations (Sleep Health Foundation, 2022). The compensatory internet use theory helps explain this pattern. While entertainment can offer short-term stress relief, excessive reliance may reinforce avoidance coping, negatively affecting wellbeing over time (Elhai et al., 2017).

The uses and gratifications theory further clarifies these risks by showing how entertainment use can shift from intentional choice to automatic checking. This diminishes autonomy and reinforces distraction habits (Sundar &amp; Limperos, 2013). Similarly, the self-determination theory suggests that hedonic use dominated by stress or obligation fails to satisfy basic needs and may undermine health (Deci &amp; Ryan, 1985).

'''Summary'''

Hedonic motivations highlights the double-edged nature of mobile entertainment. When balanced, these activities fulfil psychological needs and aid recovery from stress (Reinecke &amp; Eden, 2016). Conversely, when driven by avoidance or habit, they can impair sleep, learning, and overall wellbeing (Kuss et al., 2018). The impact depends less on the technology itself and more on whether people use it as a balanced form of enjoyment or as a substitute for healthier coping strategies.

{{RoundBoxTop|theme=9}}
[[File:Thought_bubble.svg|40px|link=]]
'''Reflection box: How do you use your phone for entertainment?'''

Think about the last time you used your phone just for enjoyment (e.g., watching videos, playing games, scrolling social media). 

*Did it help you feel more relaxed or energised?
*Did it ever interfere with your sleep?
*What needs (autonomy, competence, connection) do you think were met by that phone use?
{{RoundBoxBottom}}

== Social motivations ==
{{RoundBoxTop|theme=3}}
[[File:Girl silhouette.svg|20px|link=]]
'''Case example: Taylor'''

Taylor recently moved to a new city for university. She uses messaging apps, video calls, and social media to stay in touch with old friends. These interactions help her feel connected to her community, despite the distance.
{{RoundBoxBottom}}

Taylor’s behaviour reflects social motivations. These motivations stem from the human need for connection, belonging, and validation. Phones allow people to maintain relationships, share updates and feel included regardless of distance. Social motivations include messaging, posting content, and joining online groups. While often rewarding, social use can also create pressure to stay available and can increase stress when users feel excluded (Kim &amp; Lee, 2011).  

[[File:Online course, fall 2020, Senior Citizens Write Wikipedia.png|thumb|right|''Figure 4''. A group of friends using video calls to socialise with each other.|300x300px]]

'''Theoretical explanation'''

Three psychological models explain how social motivations drive mobile phone use.
*Self-determination theory: SDT identifies relatedness as a basic psychological need, alongside autonomy and competence (Deci &amp; Ryan, 2000). Phones support relatedness by offering frequent contact (Joo &amp; Sang, 2013). However, when connections are pursued out of obligation or fear of exclusion, the benefits of relatedness decline (Kushlev et al., 2016).
*Uses and gratifications theory: UGT sees users as active decision-makers who choose media that fulfils needs such as belonging, reassurance, or identity expression (Katz et al., 1973). Phones make this easy by providing instant connection.
*Compensatory Internet Use Theory (CIUT): CIUT proposes that people use online communication to compensate for offline social challenges (Kardefelt-Winther, 2014). For example, students may rely on group chats or social media to reduce loneliness or social anxiety. While this can provide temporary comfort, it may also reinforce dependence and reduce engagement in offline interactions.

Together, these perspectives suggest that social motivations can be both supportive and anxiety-driven.

'''Research evidence'''

Research highlights both benefits and risks associated with social phone use. On the positive side, social interactions via smartphones strengthen perceived support and bonding. Tools like messaging, [[wikipedia:Videotelephony|video calls]], and [[wikipedia:Chat_room|group chats]] help reduce loneliness and help students adapt to new environments (Joo &amp; Sang, 2013). Surveys show that sharing humour, updates, and encouragement online is linked with higher wellbeing and a stronger sense of belonging (Mission Australia, 2022). These benefits are strongest when social media use is intentional. Sundar and Limperos (2013) found that focusing on meaningful exchanges rather than constant scrolling reduces loneliness and supports resilience. From the perspective of SDT, this shows how social needs for relatedness can be fulfilled in autonomy-supportive ways (Deci &amp; Ryan, 2000).

However, risks arise when social use is excessive or driven by anxiety. [[wikipedia:Fear_of_missing_out|FoMo]] is strongly linked to compulsive checking, increased stress, and sleep disturbances (Przybylski et al., 2013). Longitudinal studies show that students who mainly use phones to alleviate social anxiety report heightened distress over time (Elhai et al., 2017). This aligns with the CIUT, which suggests that digital connection can serve as avoidance coping. It offers temporary relief but reinforces dependence (Kardefelt-Winther, 2014). UGT further explains that repeated reliance on phones for reassurance fosters habitual checking rather than active coping, increasing vulnerability to problematic use (Kuss et al., 2018).

'''Summary'''

Social motivations demonstrate how phones serve as tools for belonging. When use is selective and autonomy-supportive, phones enhance wellbeing by fulfilling relatedness needs (Deci &amp; Ryan, 2000). Conversely, when driven by FoMo or avoidance, social phone use may foster stress, dependence, and reduced offline interaction (Elhai et al., 2019). A positive balance depends on whether social phone use supports authentic connection or substitutes for healthier coping strategies.

== Emotional and mood regulatory motivations ==
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[[File:Boy from vector.svg|30px|link=]]
'''Case example: Sam'''

Sam often feels anxious before giving class presentations. To cope, he checks his phone for supportive messages and scrolls through Instagram to distract himself. This brings temporary relief but prevents him from addressing the root of his anxiety.
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Sam’s behaviour reflects emotional and mood regulatory motivations, where phones are used to manage stress, loneliness, anxiety, sadness, or boredom. Smartphones provide immediate outlets for distraction, reassurance, or self-expression. This helps stabilise mood in the short term. These motivations stem from the basic psychological drive to regulate emotions and regain a sense of control. However, reliance on phones can shift from adaptive coping to maladaptive dependence when overused.

'''Theoretical explanation'''

Three psychological models explain these dynamics.
*[[wikipedia:Emotional_self-regulation|Emotion regulation]] theory: Gross (1998) distinguishes strategies such as distraction, reappraisal, and suppression. Phones support distraction (e.g., scrolling feeds) and can encourage reappraisal by offering uplifting or humorous content.
*[[wikipedia:Mood_management_theory|Mood management theory]]: Zillmann (1988) suggests people choose media to reduce negative states or maintain positive ones. Smartphones provide instant access to entertainment, games, and social contact, making them powerful tools for mood adjustment.
*Compensatory internet use theory: Kardefelt-Winther (2014) proposes that individuals turn to media to offset offline stressors. While this can relieve tension, it may also fuel avoidance if deeper problems remain unresolved.

Together, these theories explain why emotionally motivated phone use can bring short-term relief but also carry long-term risks.

'''Research evidence'''

Moderate use of relaxation apps, mindfulness tools, or supportive messaging can reduce stress and improve wellbeing in the short term (Reinecke &amp; Eden, 2016). Social sharing and online self-expression offer [[wikipedia:Catharsis|catharsis]] and strengthen social bonds, consistent with mood management theory (Kim &amp; Lee, 2011). Additional evidence shows that reappraisal through positive online content may enhance emotional resilience (Park &amp; Valenzuela, 2009).

However, risks arise when emotional regulation relies too heavily on smartphones. Escaping stress through constant phone use has been linked to procrastination, avoidance, and increased stress (Elhai et al., 2019). Heavy nighttime use is consistently linked with poor sleep quality and heightened anxiety (Sleep Health Foundation, 2022). Longitudinal studies further show that depending on phones as a primary coping mechanism predicts worsening depressive symptoms over time, especially in adolescents (Kuss et al., 2018). These findings illustrate the shift from adaptive distraction to maladaptive dependence when stressors remain unaddressed.

[[File:Depiction of a person suffering from Insomnia (sleeplessness) (cropped).png |thumb|''Figure 5.'' Illustration of a woman struggling to fall asleep at night because she is using her phone before bed.]]'''Summary'''

Emotion regulation theory highlights how phones provide quick access to distraction and reappraisal. Mood management theory explains why people seek uplifting or entertaining content. Compensatory internet use theory clarifies why these strategies sometimes slip into avoidance and dependency. Using digital tools is most beneficial when they complement healthy offline coping strategies such as problem-solving, rest, and social support, rather than replacing them (Kushlev et al., 2016).

== Practical implications ==
To promote healthier smartphone use, strategies must address the different motivations that drive behaviour. Grounding these strategies in psychological theory ensures that they move beyond surface-level advice and instead build lasting skills for wellbeing. By tailoring strategies, smartphone use can shift from automatic or avoidant behaviours to intentional habits that enhance autonomy, competence, and relatedness (Deci &amp; Ryan, 2000).

'''Instrumental use'''

Smartphones are valuable tools for organisation, navigation, and learning. Using calendar apps, focus timers, and note-taking platforms can strengthen autonomy and competence (Deci &amp; Ryan, 2000). Practical strategies could include driving offline routes to maintain spatial awareness while reducing an over-reliance on devices like GPS (Ruginski et al., 2019). Also, reflecting on which apps genuinely support goals helps build self-regulation (Hadlington, 2015). Workplaces, schools, and families can also encourage digital literacy training to help people use apps more intentionally, aligning with self-regulation theory, which emphasises feedback and planning in achieving goals.

'''Hedonic use'''

Phones provide entertainment and while these activities can improve mood, excessive use can disrupt sleep and concentration (Exelmans &amp; Van den Bulck, 2016). Practical strategies include setting app timers, scheduling intentional leisure breaks, and prioritising stimulating media over passive media (Sundar &amp; Limperos, 2013). Choosing creativity-focused or interactive content enhances wellbeing. Also reducing device use before sleep supports circadian rhythms and reduces fatigue (Exelmans &amp; Van den Bulck, 2016). From the perspective of mood management theory (Zillmann, 1988), these strategies work by channelling hedonic needs into intentional rather than automatic behaviours. This allows people to enjoy pleasure while minimising costs to health and productivity.

'''Social use'''

Connection with others is a central reason for phone use. Structured strategies, such as curated group chats, online communities, or family check-ins, allow people to maintain relationships without constant checking. Intentional social use through prioritising direct communication over passive scrolling strengthens bonds and wellbeing (Beyens et al., 2020). Reducing notifications or batching social media checks can lower compulsive behaviours (Przybylski et al., 2013). Reflecting on whether engagement arises from genuine interest or obligation helps protect autonomy in relationships (Kushlev et al., 2016). Compensatory Internet Use Theory (Kardefelt-Winther, 2014) suggests that many turn to digital connection to offset stress or loneliness. Pairing online interactions with offline opportunities for belonging, such as community groups or shared hobbies, can reduce dependence while still supporting relatedness.

[[File:A young girl listening to music on her phone.png|thumb|''Figure 6.'' An illustration of a young girl listening to music on her mobile phone.]]

'''Emotional use'''

Smartphones are often used to cope with stress, sadness, or boredom. Adaptive strategies include guided relaxation apps, calming audio, or mood-tracking tools, which can reduce stress in the short term (Reinecke &amp; Eden, 2016). However, coping is most effective when digital tools are paired with offline support such as rest, social support, or problem-solving (Gross, 1998). Reflective practices such as asking “Is this helping me cope, or am I avoiding?” can prevent avoidance cycles linked with poor sleep and heightened anxiety (Elhai et al., 2017). Mood management theory (Zillmann, 1988) {{ic|Not in References}} explains why uplifting content helps regulate emotions. While compensatory internet use theory highlights the risks of relying on avoidance strategies without addressing underlying stressors.

==Conclusion==
Smartphone use is driven by instrumental, hedonic, social, and emotional motivations, reflecting basic psychological needs. Psychological theories like TAM, SDT, and UGT show that phones are not inherently good or bad. Their impact depends on the motivations guiding engagement. Smartphones can support productivity, enjoyment, connection, and stress relief, but the same patterns may also foster avoidance, dependency, and reduced wellbeing. This dual role highlights the importance of intentional use. Hedonic use is most rewarding when balanced, and social connection most beneficial when authentic. Instrumental and emotional use require reflection to avoid over-reliance. Recognising personal motivations helps individuals use smartphones as tools for growth, protecting autonomy, rest, and wellbeing. By understanding these dynamics, people can shape technology into an ally for resilience rather than a source of distraction.

==See also==
* [[Motivation and emotion/Book/2021/Boredom and technology addiction|Boredom and technology addiction]] (Book chapter, 2021)
* [[Motivation and emotion/Book/2016/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2016)
* [[Motivation and emotion/Book/2017/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2017) 
* [[Motivation and emotion/Book/2023/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2023)

==References==
{{Hanging indent|1=

Beyens, I., Pouwels, J. L., van Driel, I. I., Keijsers, L., &amp; Valkenburg, P. M. (2020). The effect of social media on well-being differs from adolescent to adolescent. ''Scientific Reports'', ''10'', 10763. https://doi.org/10.1038/s41598-020-67727-7

Davis, F. D. (1989). Perceived usefulness, perceived ease of use, and user acceptance of information technology. ''MIS Quarterly'', ''13''(3), 319–340. https://doi.org/10.2307/249008

Deci, E. L., &amp; Ryan, R. M. (1985). ''Intrinsic motivation and self-determination in human behavior''. Plenum Press. https://doi.org/10.1007/978-1-4899-2271-7

Deci, E. L., &amp; Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. ''Psychological Inquiry'', ''11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Elhai, J. D., Levine, J. C., Dvorak, R. D., &amp; Hall, B. J. (2017). Non-social features of smartphone use are most related to depression, anxiety and problematic smartphone use. ''Computers in Human Behavior'', ''69'', 75–82. https://doi.org/10.1016/j.chb.2016.12.023

Elhai, J. D., Yang, H., McKay, D., &amp; Asmundson, G. J. (2019). Depression and anxiety symptoms are related to problematic smartphone use severity in Chinese young adults: Fear of missing out as a mediator. ''Addictive Behaviors'', ''101'', 105962. https://doi.org/10.1016/j.addbeh.2019.04.020

Exelmans, L., &amp; Van den Bulck, J. (2016). Bedtime mobile phone use and sleep in adults. ''Social Science &amp; Medicine'', ''148'', 93–101. https://doi.org/10.1016/j.socscimed.2015.11.037

Fishbein, M., &amp; Ajzen, I. (1975). ''Belief, attitude, intention, and behavior: An introduction to theory and research''. Addison-Wesley. https://people.umass.edu/aizen/f&amp;a1975.html

Granic, I., Lobel, A., &amp; Engels, R. C. M. E. (2014). The benefits of playing video games. ''American Psychologist'', ''69''(1), 66–78. https://doi.org/10.1037/a0034857

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/10.1037/1089-2680.2.3.271

Hadlington, L. (2015). Cognitive failures in daily life: Exploring the link with Internet addiction and problematic mobile phone use. ''Computers in Human Behavior'', ''51'', 75–81. https://doi.org/10.1016/j.chb.2015.04.036

Joo, T. M., &amp; Sang, Y. (2013). Exploring Koreans’ smartphone usage: An integrated model of the technology acceptance model and uses and gratifications theory. ''Computers in Human Behavior'', ''29''(6), 2512–2518. https://doi.org/10.1016/j.chb.2013.06.002

Kardefelt-Winther, D. (2014). A conceptual and methodological critique of Internet addiction research: Towards a model of compensatory Internet use. ''Computers in Human Behavior'', ''31'', 351–354. https://doi.org/10.1016/j.chb.2013.10.059

Katz, E., Blumler, J. G., &amp; Gurevitch, M. (1973). Uses and gratifications research. ''Public Opinion Quarterly'', ''37''(4), 509–523. https://doi.org/10.1086/268109

Kim, J., &amp; Lee, J. E. R. (2011). The Facebook paths to happiness: Effects of the number of Facebook friends and self-presentation on subjective well-being. ''Cyberpsychology, Behavior, and Social Networking'', ''14''(6), 359–364. https://doi.org/10.1089/cyber.2010.0374

Kushlev, K., Proulx, J., &amp; Dunn, E. W. (2016). Digitally connected, socially disconnected: The effects of relying on technology rather than other people. ''Computers in Human Behavior'', ''58'', 140–148. https://doi.org/10.1016/j.chb.2017.07.001

Kuss, D. J., Griffiths, M. D., Karila, L., &amp; Billieux, J. (2018). Internet addiction: A systematic review of epidemiological research for the last decade. ''Current Pharmaceutical Design'', ''20''(25), 4026–4052. https://doi.org/10.2174/13816128113199990617

Meng, H., Cao, H., Hao, R., Zhou, N., Liang, Y., Wu, L., Jiang, L., Ma, R., Li, B., Deng, L., Lin, Z., Lin, X., &amp; Zhang, J. (2020). Smartphone use motivation and problematic smartphone use in a national representative sample of Chinese adolescents: The mediating roles of smartphone use time for various activities. ''Journal of behavioral addictions'', ''9''(1), 163–174. https://doi.org/10.1556/2006.2020.00004

Park, N., Kee, K. F., &amp; Valenzuela, S. (2009). Being immersed in social networking environment: Facebook groups, uses and gratifications, and social outcomes. ''Cyberpsychology, Behavior, and Social Networking'', ''13''(6), 357–360. https://doi.org/10.1089/cpb.2009.0003

Przybylski, A. K., Murayama, K., DeHaan, C. R., &amp; Gladwell, V. (2013). Motivational, emotional, and behavioral correlates of fear of missing out. ''Computers in Human Behavior'', ''29''(4), 1841–1848. https://doi.org/10.1016/j.chb.2013.02.014

Reinecke, L., &amp; Eden, A. (2016). Media use and recreation: Media-induced recovery as a link between media exposure and well-being. In L. Reinecke &amp; M. B. Oliver (Eds.), ''The Routledge handbook of media use and well-being'' (pp. 106–117). Routledge. https://doi.org/10.4324/9781315714752

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., &amp; Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. ''Journal of Environmental Psychology'', ''64'', 12–20. https://doi.org/10.1016/j.jenvp.2019.05.001

Sundar, S. S., &amp; Limperos, A. M. (2013). Uses and grats 2.0: New gratifications for new media. ''Journal of Broadcasting &amp; Electronic Media'', ''57''(4), 504–525. https://doi.org/10.1080/08838151.2013.845827

Wilson, S. A., Byrne, P., Rodgers, S. E., &amp; Maden, M. (2022). A systematic review of smartphone and tablet use by older adults with and without cognitive impairment. ''Innovation in Aging'', ''6''(2), igac002. https://doi.org/10.1093/geroni/igac002

}}

==External links==
* [https://www.sleephealthfoundation.org.au/sleep-topics/technology-and-sleep Technology and sleep] (Sleep Health Foundation)
* [https://www.missionaustralia.com.au/publications/youth-survey/state-reports-2022?layout=columns Youth survey report 2022] (Mission Australia)

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== Overview ==
{{RoundBoxTop|theme=3}}
[[File:M100 peruskorjattu sisältä.jpg|thumb|''' ''Figure 1''.''' People using their phone on the train.]]

'''Imagine this ...'''

You are sitting on a crowded train after a long day. A young boy nearby laughs at a meme on Instagram. A woman beside him scrolls through her messages. Across the aisle, a student types reminders into a calendar app. Next to you, a man endlessly scrolls through social media updates. What draws each of these people to their phone? Is it comfort, connection, distraction, or control? These are everyday actions, yet behind them lie hidden motivations that shape our relationship with smartphones.
{{RoundBoxBottom}}

[[wikipedia:Mobile_phone|Mobile phones]] are central to modern life, influencing how people work, connect, and cope with [[wikipedia:Stress_(biology)|stress]]. Their use is driven by complex psychological needs that affect daily [[wikipedia:Behavior|behaviour]] and [[wikipedia:Well-being|wellbeing]]. Understanding these [[wikipedia:Motivation|motivations]] reveals why people engage with phones in unique ways, and why these behaviours can bring both benefits and challenges. [[wikipedia:Smartphone|Smartphones]] enhance productivity, entertainment, and social bonds. Yet, excessive or compulsive use can lead to [[wikipedia:Procrastination|procrastination]], [[wikipedia:Anxiety|anxiety]], and emotional [[wikipedia:Avoidance_coping|avoidance]]. This chapter explores how motivations shape phone use and offers strategies for healthier, intentional smartphone use that supports wellbeing.

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What motivates people to use smartphones in their daily lives?
* What psychological theories explain these motivations?
* How do different motivations impact wellbeing in positive and negative ways?
* What strategies can encourage mindful and beneficial phone use?
{{RoundBoxBottom}}

== Key motivations for mobile phone use ==
People use mobile phones to satisfy different psychological needs. These needs are typically categorised as instrumental (task-focused), hedonic (pleasure-oriented), social (connection-based), and emotional (coping-related) (Sundar &amp; Limperos, 2013). These motivations explain how phones are integrated into daily life and how they influence wellbeing (Davis, 1989).

== Instrumental and pragmatic motivations ==
{{RoundBoxTop|theme=1}}[[File:Boy from vector.svg|30px|link=]] '''Case example: Alex'''

Alex starts each morning by checking the weather app, navigating traffic with GPS, reviewing lecture notes, and setting assignments reminders. Each action is purposeful and task-oriented. His smartphone works as a practical assistant, helping him manage academic and daily tasks.
{{RoundBoxBottom}}

Instrumental motivations describe using phones to achieve specific goals. These motivations support productivity, scheduling, navigation, and information access (Joo &amp; Sang, 2013). Pragmatic motivations are a subtype of instrumental motivations, focusing on immediate benefits such as reassurance during emergencies or quick information access (Meng et al., 2020). Together, these motivations position smartphones as essential tools for managing everyday life (Wilson et al., 2022).

[[File:Online calendar used for scheduling.png|thumb|''Figure 2.'' An example of an online calendar used daily to organise and schedule tasks.|300x300px]]'''Theoretical explanation'''

Two psychological models explain these motivations.
*The [[wikipedia:Technology_acceptance_model|technology acceptance model]] (TAM): TAM proposes that technology use depends on perceived usefulness and ease of use (Davis, 1989). When phones are both effective and effortless, they are more likely to be integrated into daily routines. This aligns with the [[wikipedia:Theory_of_reasoned_action|theory of reasoned action]], which views attitudes as strong predictors of behaviour (Fishbein &amp; Ajzen, 1975).
*The [[wikipedia:Uses_and_gratifications_theory|uses and gratifications theory]] (UGT): UGT views smartphone users as active decision-makers who adopt media to fulfil personal needs (Katz et al., 1973). Instrumental use reflects utility-based gratifications, such as finding directions, setting reminders, or contacting help (Park et al., 2009).

These models show how technology design and user choice shape instrumental motivation.

'''Research evidence'''

Instrumental phone use is consistently linked to positive outcomes. Purposeful engagement such as accessing information, improving productivity, and navigating with GPS supports daily functioning (Wilson et al., 2022). For older adults, instrumental use helps coordinate activities and supports both cognitive and social functioning (Wilson et al., 2022). Pragmatic motivations extend these benefits by providing reassurance and safety, especially during emergencies (Meng et al., 2020). From TAM's perspective, these outcomes illustrate how perceived usefulness encourages routine phone behaviour (Davis, 1989).

Meng et al. (2020) found that instrumental use is less likely to lead to dependency compared to hedonic or emotional motivations. This suggests instrumental use promotes a balanced engagement with smartphones. However, risks exist. An over-reliance on GPS may impair spatial awareness (Ruginski et al., 2019), and habitual checking of reminders can weaken memory and problem-solving skills (Elhai et al., 2017). UGT helps explain this shift from adaptive to maladaptive outcomes. While users initially select tools for convenience, {{g]} however reliance can turn into habit, diminishing autonomy and flexibility (Kardefelt-Winther, 2014).

'''Summary'''

Instrumental and pragmatic motivations generally support adaptive outcomes. Phones act as practical assistants, helping people stay organised, informed, and safe. TAM and UGT show that when phones are useful and easy to use, they integrate smoothly into daily life. Yet, the same convenience may reduce independence if it replaces critical skills. These motivations are most beneficial when phones complement rather than replace human capabilities (Joo &amp; Sang, 2013).

== Hedonic and entertainment motivations ==
{{RoundBoxTop|theme=2}}[[File:Girl silhouette.svg|20px|link=]] '''Case example: Sarah'''

After a long day of studying, Sarah relaxes by watching videos on TikTok or playing mobile games like Clash of Clans. These activities improve her mood and distract her from her current stress. But sometimes these activities delay her sleep and cut into her study time.
{{RoundBoxBottom}}

[[File:Young girl Using Her Phone.jpg|thumb|''Figure 3.'' A young woman engaged in using her mobile phone.]]

Hedonic motivations describe using phones for enjoyment, escape, and relief from boredom. This includes watching videos, gaming, scrolling [[wikipedia:Social_media|social media]], or listening to music. These activities provide quick pleasure and immersion (Granic et al., 2014).    
'''Theoretical explanation'''

Three psychological perspectives explain how hedonic motivations drive mobile phone use.
*The uses and gratifications theory: People actively select media to meet needs such as entertainment, diversion, and relaxation (Katz et al., 1973). Smartphones amplify these gratifications because they combine many entertainment functions in one portable device (Sundar &amp; Limperos, 2013).
*[[wikipedia:Self-determination_theory|Self-determination theory (SDT)]]: Hedonic use can satisfy the basic psychological needs for autonomy and competence, enhancing intrinsic motivation (Deci &amp; Ryan, 1985) For example, choosing a playlist supports autonomy. Beating a level in a game provides competence. Sharing a video with friends strengthens relatedness. When these needs are met, hedonic use can enhance wellbeing (Jeno et al., 2017).
*The compensatory internet use theory (CIUT): People use online entertainment to cope with stress, loneliness, or unmet needs (Kardefelt-Winther, 2014). For students like Sarah, this may mean escaping academic pressure through binge-watching or gaming. While this coping strategy may provide temporary relief, it can also become maladaptive if it replaces healthier habits like rest or social interaction (Elhai et al., 2019).

Together, these theories show that hedonic use is about mood regulation, need fulfilment, and distraction.

'''Research evidence'''

Short bursts of gaming or video watching can reduce stress and help recovery from daily strain (Reinecke &amp; Eden, 2016). Adolescents who engage in these activities with peers also report stronger friendships and an increased sense of belonging (Granic et al., 2014). From the perspective of self-determination theory, these hedonic activities satisfy psychological needs for competence and relatedness, supporting wellbeing when used in moderation (Deci &amp; Ryan, 2000).

However, risks arise when hedonic use becomes excessive. A meta-analysis by Kuss et al. (2018) found a moderate association between hedonic phone use and poorer sleep quality. This corroborates findings from national survey data. For instance, 70% of Australian teenagers use their phones in bed, with half of them reporting shorter sleep durations (Sleep Health Foundation, 2022). The compensatory internet use theory helps explain this pattern. While entertainment can offer short-term stress relief, excessive reliance may reinforce avoidance coping, negatively affecting wellbeing over time (Elhai et al., 2017).

The uses and gratifications theory further clarifies these risks by showing how entertainment use can shift from intentional choice to automatic checking. This diminishes autonomy and reinforces distraction habits (Sundar &amp; Limperos, 2013). Similarly, the self-determination theory suggests that hedonic use dominated by stress or obligation fails to satisfy basic needs and may undermine health (Deci &amp; Ryan, 1985).

'''Summary'''

Hedonic motivations highlights the double-edged nature of mobile entertainment. When balanced, these activities fulfil psychological needs and aid recovery from stress (Reinecke &amp; Eden, 2016). Conversely, when driven by avoidance or habit, they can impair sleep, learning, and overall wellbeing (Kuss et al., 2018). The impact depends less on the technology itself and more on whether people use it as a balanced form of enjoyment or as a substitute for healthier coping strategies.

{{RoundBoxTop|theme=9}}
[[File:Thought_bubble.svg|40px|link=]]
'''Reflection box: How do you use your phone for entertainment?'''

Think about the last time you used your phone just for enjoyment (e.g., watching videos, playing games, scrolling social media). 

*Did it help you feel more relaxed or energised?
*Did it ever interfere with your sleep?
*What needs (autonomy, competence, connection) do you think were met by that phone use?
{{RoundBoxBottom}}

== Social motivations ==
{{RoundBoxTop|theme=3}}
[[File:Girl silhouette.svg|20px|link=]]
'''Case example: Taylor'''

Taylor recently moved to a new city for university. She uses messaging apps, video calls, and social media to stay in touch with old friends. These interactions help her feel connected to her community, despite the distance.
{{RoundBoxBottom}}

Taylor’s behaviour reflects social motivations. These motivations stem from the human need for connection, belonging, and validation. Phones allow people to maintain relationships, share updates and feel included regardless of distance. Social motivations include messaging, posting content, and joining online groups. While often rewarding, social use can also create pressure to stay available and can increase stress when users feel excluded (Kim &amp; Lee, 2011).  

[[File:Online course, fall 2020, Senior Citizens Write Wikipedia.png|thumb|right|''Figure 4''. A group of friends using video calls to socialise with each other.|300x300px]]

'''Theoretical explanation'''

Three psychological models explain how social motivations drive mobile phone use.
*Self-determination theory: SDT identifies relatedness as a basic psychological need, alongside autonomy and competence (Deci &amp; Ryan, 2000). Phones support relatedness by offering frequent contact (Joo &amp; Sang, 2013). However, when connections are pursued out of obligation or fear of exclusion, the benefits of relatedness decline (Kushlev et al., 2016).
*Uses and gratifications theory: UGT sees users as active decision-makers who choose media that fulfils needs such as belonging, reassurance, or identity expression (Katz et al., 1973). Phones make this easy by providing instant connection.
*Compensatory Internet Use Theory (CIUT): CIUT proposes that people use online communication to compensate for offline social challenges (Kardefelt-Winther, 2014). For example, students may rely on group chats or social media to reduce loneliness or social anxiety. While this can provide temporary comfort, it may also reinforce dependence and reduce engagement in offline interactions.

Together, these perspectives suggest that social motivations can be both supportive and anxiety-driven.

'''Research evidence'''

Research highlights both benefits and risks associated with social phone use. On the positive side, social interactions via smartphones strengthen perceived support and bonding. Tools like messaging, [[wikipedia:Videotelephony|video calls]], and [[wikipedia:Chat_room|group chats]] help reduce loneliness and help students adapt to new environments (Joo &amp; Sang, 2013). Surveys show that sharing humour, updates, and encouragement online is linked with higher wellbeing and a stronger sense of belonging (Mission Australia, 2022). These benefits are strongest when social media use is intentional. Sundar and Limperos (2013) found that focusing on meaningful exchanges rather than constant scrolling reduces loneliness and supports resilience. From the perspective of SDT, this shows how social needs for relatedness can be fulfilled in autonomy-supportive ways (Deci &amp; Ryan, 2000).

However, risks arise when social use is excessive or driven by anxiety. [[wikipedia:Fear_of_missing_out|FoMo]] is strongly linked to compulsive checking, increased stress, and sleep disturbances (Przybylski et al., 2013). Longitudinal studies show that students who mainly use phones to alleviate social anxiety report heightened distress over time (Elhai et al., 2017). This aligns with the CIUT, which suggests that digital connection can serve as avoidance coping. It offers temporary relief but reinforces dependence (Kardefelt-Winther, 2014). UGT further explains that repeated reliance on phones for reassurance fosters habitual checking rather than active coping, increasing vulnerability to problematic use (Kuss et al., 2018).

'''Summary'''

Social motivations demonstrate how phones serve as tools for belonging. When use is selective and autonomy-supportive, phones enhance wellbeing by fulfilling relatedness needs (Deci &amp; Ryan, 2000). Conversely, when driven by FoMo or avoidance, social phone use may foster stress, dependence, and reduced offline interaction (Elhai et al., 2019). A positive balance depends on whether social phone use supports authentic connection or substitutes for healthier coping strategies.

== Emotional and mood regulatory motivations ==
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[[File:Boy from vector.svg|30px|link=]]
'''Case example: Sam'''

Sam often feels anxious before giving class presentations. To cope, he checks his phone for supportive messages and scrolls through Instagram to distract himself. This brings temporary relief but prevents him from addressing the root of his anxiety.
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Sam’s behaviour reflects emotional and mood regulatory motivations, where phones are used to manage stress, loneliness, anxiety, sadness, or boredom. Smartphones provide immediate outlets for distraction, reassurance, or self-expression. This helps stabilise mood in the short term. These motivations stem from the basic psychological drive to regulate emotions and regain a sense of control. However, reliance on phones can shift from adaptive coping to maladaptive dependence when overused.

'''Theoretical explanation'''

Three psychological models explain these dynamics.
*[[wikipedia:Emotional_self-regulation|Emotion regulation]] theory: Gross (1998) distinguishes strategies such as distraction, reappraisal, and suppression. Phones support distraction (e.g., scrolling feeds) and can encourage reappraisal by offering uplifting or humorous content.
*[[wikipedia:Mood_management_theory|Mood management theory]]: Zillmann (1988) suggests people choose media to reduce negative states or maintain positive ones. Smartphones provide instant access to entertainment, games, and social contact, making them powerful tools for mood adjustment.
*Compensatory internet use theory: Kardefelt-Winther (2014) proposes that individuals turn to media to offset offline stressors. While this can relieve tension, it may also fuel avoidance if deeper problems remain unresolved.

Together, these theories explain why emotionally motivated phone use can bring short-term relief but also carry long-term risks.

'''Research evidence'''

Moderate use of relaxation apps, mindfulness tools, or supportive messaging can reduce stress and improve wellbeing in the short term (Reinecke &amp; Eden, 2016). Social sharing and online self-expression offer [[wikipedia:Catharsis|catharsis]] and strengthen social bonds, consistent with mood management theory (Kim &amp; Lee, 2011). Additional evidence shows that reappraisal through positive online content may enhance emotional resilience (Park &amp; Valenzuela, 2009).

However, risks arise when emotional regulation relies too heavily on smartphones. Escaping stress through constant phone use has been linked to procrastination, avoidance, and increased stress (Elhai et al., 2019). Heavy nighttime use is consistently linked with poor sleep quality and heightened anxiety (Sleep Health Foundation, 2022). Longitudinal studies further show that depending on phones as a primary coping mechanism predicts worsening depressive symptoms over time, especially in adolescents (Kuss et al., 2018). These findings illustrate the shift from adaptive distraction to maladaptive dependence when stressors remain unaddressed.

[[File:Depiction of a person suffering from Insomnia (sleeplessness) (cropped).png |thumb|''Figure 5.'' Illustration of a woman struggling to fall asleep at night because she is using her phone before bed.]]'''Summary'''

Emotion regulation theory highlights how phones provide quick access to distraction and reappraisal. Mood management theory explains why people seek uplifting or entertaining content. Compensatory internet use theory clarifies why these strategies sometimes slip into avoidance and dependency. Using digital tools is most beneficial when they complement healthy offline coping strategies such as problem-solving, rest, and social support, rather than replacing them (Kushlev et al., 2016).

== Practical implications ==
To promote healthier smartphone use, strategies must address the different motivations that drive behaviour. Grounding these strategies in psychological theory ensures that they move beyond surface-level advice and instead build lasting skills for wellbeing. By tailoring strategies, smartphone use can shift from automatic or avoidant behaviours to intentional habits that enhance autonomy, competence, and relatedness (Deci &amp; Ryan, 2000).

'''Instrumental use'''

Smartphones are valuable tools for organisation, navigation, and learning. Using calendar apps, focus timers, and note-taking platforms can strengthen autonomy and competence (Deci &amp; Ryan, 2000). Practical strategies could include driving offline routes to maintain spatial awareness while reducing an over-reliance on devices like GPS (Ruginski et al., 2019). Also, reflecting on which apps genuinely support goals helps build self-regulation (Hadlington, 2015). Workplaces, schools, and families can also encourage digital literacy training to help people use apps more intentionally, aligning with self-regulation theory, which emphasises feedback and planning in achieving goals.

'''Hedonic use'''

Phones provide entertainment and while these activities can improve mood, excessive use can disrupt sleep and concentration (Exelmans &amp; Van den Bulck, 2016). Practical strategies include setting app timers, scheduling intentional leisure breaks, and prioritising stimulating media over passive media (Sundar &amp; Limperos, 2013). Choosing creativity-focused or interactive content enhances wellbeing. Also reducing device use before sleep supports circadian rhythms and reduces fatigue (Exelmans &amp; Van den Bulck, 2016). From the perspective of mood management theory (Zillmann, 1988), these strategies work by channelling hedonic needs into intentional rather than automatic behaviours. This allows people to enjoy pleasure while minimising costs to health and productivity.

'''Social use'''

Connection with others is a central reason for phone use. Structured strategies, such as curated group chats, online communities, or family check-ins, allow people to maintain relationships without constant checking. Intentional social use through prioritising direct communication over passive scrolling strengthens bonds and wellbeing (Beyens et al., 2020). Reducing notifications or batching social media checks can lower compulsive behaviours (Przybylski et al., 2013). Reflecting on whether engagement arises from genuine interest or obligation helps protect autonomy in relationships (Kushlev et al., 2016). Compensatory Internet Use Theory (Kardefelt-Winther, 2014) suggests that many turn to digital connection to offset stress or loneliness. Pairing online interactions with offline opportunities for belonging, such as community groups or shared hobbies, can reduce dependence while still supporting relatedness.

[[File:A young girl listening to music on her phone.png|thumb|''Figure 6.'' An illustration of a girl listening to music on her mobile phone.]]

'''Emotional use'''

Smartphones are often used to cope with stress, sadness, or boredom. Adaptive strategies include guided relaxation apps, calming audio, or mood-tracking tools, which can reduce stress in the short term (Reinecke &amp; Eden, 2016). However, coping is most effective when digital tools are paired with offline support such as rest, social support, or problem-solving (Gross, 1998). Reflective practices such as asking “Is this helping me cope, or am I avoiding?” can prevent avoidance cycles linked with poor sleep and heightened anxiety (Elhai et al., 2017). Mood management theory (Zillmann, 1988) {{ic|Not in References}} explains why uplifting content helps regulate emotions. While compensatory internet use theory highlights the risks of relying on avoidance strategies without addressing underlying stressors.

==Conclusion==
Smartphone use is driven by instrumental, hedonic, social, and emotional motivations, reflecting basic psychological needs. Psychological theories like TAM, SDT, and UGT show that phones are not inherently good or bad. Their impact depends on the motivations guiding engagement. Smartphones can support productivity, enjoyment, connection, and stress relief, but the same patterns may also foster avoidance, dependency, and reduced wellbeing. This dual role highlights the importance of intentional use. Hedonic use is most rewarding when balanced, and social connection most beneficial when authentic. Instrumental and emotional use require reflection to avoid over-reliance. Recognising personal motivations helps individuals use smartphones as tools for growth, protecting autonomy, rest, and wellbeing. By understanding these dynamics, people can shape technology into an ally for resilience rather than a source of distraction.

==See also==
* [[Motivation and emotion/Book/2021/Boredom and technology addiction|Boredom and technology addiction]] (Book chapter, 2021)
* [[Motivation and emotion/Book/2016/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2016)
* [[Motivation and emotion/Book/2017/Mobile phone addiction|Mobile phone addiction]] (Book chapter, 2017) 
* [[Motivation and emotion/Book/2023/Mobile phone use motivation|Mobile phone use motivation]] (Book chapter, 2023)

==References==
{{Hanging indent|1=

Beyens, I., Pouwels, J. L., van Driel, I. I., Keijsers, L., &amp; Valkenburg, P. M. (2020). The effect of social media on well-being differs from adolescent to adolescent. ''Scientific Reports'', ''10'', 10763. https://doi.org/10.1038/s41598-020-67727-7

Davis, F. D. (1989). Perceived usefulness, perceived ease of use, and user acceptance of information technology. ''MIS Quarterly'', ''13''(3), 319–340. https://doi.org/10.2307/249008

Deci, E. L., &amp; Ryan, R. M. (1985). ''Intrinsic motivation and self-determination in human behavior''. Plenum Press. https://doi.org/10.1007/978-1-4899-2271-7

Deci, E. L., &amp; Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. ''Psychological Inquiry'', ''11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Elhai, J. D., Levine, J. C., Dvorak, R. D., &amp; Hall, B. J. (2017). Non-social features of smartphone use are most related to depression, anxiety and problematic smartphone use. ''Computers in Human Behavior'', ''69'', 75–82. https://doi.org/10.1016/j.chb.2016.12.023

Elhai, J. D., Yang, H., McKay, D., &amp; Asmundson, G. J. (2019). Depression and anxiety symptoms are related to problematic smartphone use severity in Chinese young adults: Fear of missing out as a mediator. ''Addictive Behaviors'', ''101'', 105962. https://doi.org/10.1016/j.addbeh.2019.04.020

Exelmans, L., &amp; Van den Bulck, J. (2016). Bedtime mobile phone use and sleep in adults. ''Social Science &amp; Medicine'', ''148'', 93–101. https://doi.org/10.1016/j.socscimed.2015.11.037

Fishbein, M., &amp; Ajzen, I. (1975). ''Belief, attitude, intention, and behavior: An introduction to theory and research''. Addison-Wesley. https://people.umass.edu/aizen/f&amp;a1975.html

Granic, I., Lobel, A., &amp; Engels, R. C. M. E. (2014). The benefits of playing video games. ''American Psychologist'', ''69''(1), 66–78. https://doi.org/10.1037/a0034857

Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. https://doi.org/10.1037/1089-2680.2.3.271

Hadlington, L. (2015). Cognitive failures in daily life: Exploring the link with Internet addiction and problematic mobile phone use. ''Computers in Human Behavior'', ''51'', 75–81. https://doi.org/10.1016/j.chb.2015.04.036

Joo, T. M., &amp; Sang, Y. (2013). Exploring Koreans’ smartphone usage: An integrated model of the technology acceptance model and uses and gratifications theory. ''Computers in Human Behavior'', ''29''(6), 2512–2518. https://doi.org/10.1016/j.chb.2013.06.002

Kardefelt-Winther, D. (2014). A conceptual and methodological critique of Internet addiction research: Towards a model of compensatory Internet use. ''Computers in Human Behavior'', ''31'', 351–354. https://doi.org/10.1016/j.chb.2013.10.059

Katz, E., Blumler, J. G., &amp; Gurevitch, M. (1973). Uses and gratifications research. ''Public Opinion Quarterly'', ''37''(4), 509–523. https://doi.org/10.1086/268109

Kim, J., &amp; Lee, J. E. R. (2011). The Facebook paths to happiness: Effects of the number of Facebook friends and self-presentation on subjective well-being. ''Cyberpsychology, Behavior, and Social Networking'', ''14''(6), 359–364. https://doi.org/10.1089/cyber.2010.0374

Kushlev, K., Proulx, J., &amp; Dunn, E. W. (2016). Digitally connected, socially disconnected: The effects of relying on technology rather than other people. ''Computers in Human Behavior'', ''58'', 140–148. https://doi.org/10.1016/j.chb.2017.07.001

Kuss, D. J., Griffiths, M. D., Karila, L., &amp; Billieux, J. (2018). Internet addiction: A systematic review of epidemiological research for the last decade. ''Current Pharmaceutical Design'', ''20''(25), 4026–4052. https://doi.org/10.2174/13816128113199990617

Meng, H., Cao, H., Hao, R., Zhou, N., Liang, Y., Wu, L., Jiang, L., Ma, R., Li, B., Deng, L., Lin, Z., Lin, X., &amp; Zhang, J. (2020). Smartphone use motivation and problematic smartphone use in a national representative sample of Chinese adolescents: The mediating roles of smartphone use time for various activities. ''Journal of behavioral addictions'', ''9''(1), 163–174. https://doi.org/10.1556/2006.2020.00004

Park, N., Kee, K. F., &amp; Valenzuela, S. (2009). Being immersed in social networking environment: Facebook groups, uses and gratifications, and social outcomes. ''Cyberpsychology, Behavior, and Social Networking'', ''13''(6), 357–360. https://doi.org/10.1089/cpb.2009.0003

Przybylski, A. K., Murayama, K., DeHaan, C. R., &amp; Gladwell, V. (2013). Motivational, emotional, and behavioral correlates of fear of missing out. ''Computers in Human Behavior'', ''29''(4), 1841–1848. https://doi.org/10.1016/j.chb.2013.02.014

Reinecke, L., &amp; Eden, A. (2016). Media use and recreation: Media-induced recovery as a link between media exposure and well-being. In L. Reinecke &amp; M. B. Oliver (Eds.), ''The Routledge handbook of media use and well-being'' (pp. 106–117). Routledge. https://doi.org/10.4324/9781315714752

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., &amp; Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. ''Journal of Environmental Psychology'', ''64'', 12–20. https://doi.org/10.1016/j.jenvp.2019.05.001

Sundar, S. S., &amp; Limperos, A. M. (2013). Uses and grats 2.0: New gratifications for new media. ''Journal of Broadcasting &amp; Electronic Media'', ''57''(4), 504–525. https://doi.org/10.1080/08838151.2013.845827

Wilson, S. A., Byrne, P., Rodgers, S. E., &amp; Maden, M. (2022). A systematic review of smartphone and tablet use by older adults with and without cognitive impairment. ''Innovation in Aging'', ''6''(2), igac002. https://doi.org/10.1093/geroni/igac002

}}

==External links==
* [https://www.sleephealthfoundation.org.au/sleep-topics/technology-and-sleep Technology and sleep] (Sleep Health Foundation)
* [https://www.missionaustralia.com.au/publications/youth-survey/state-reports-2022?layout=columns Youth survey report 2022] (Mission Australia)

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[[Category:Motivation and emotion/Book/Motivation]]</text>
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Whisky (whiskey) is a type of liquor made from fermented grain mash. Various grains are used for different varieties, including barley, corn, rye, and wheat. Whisky is typically aged in charred wooden oak casks.
==Names==
There is two ways you can spell whisky. It's whisky or whiskey. The spelling whiskey is common in Ireland and the United States, while whisky is used in all other whisky-producing countries. Whisky made in Scotland is simply called whisky within Scotland. But it is also common to be called scotch(especially in north America). 
==History==
[[File:Whisky Barrels .jpg|150px|right|]]
When whisky was invented, it was way different from the whisky we know nowadays. Originally, it was not aged in oak casks, so it was very raw and strong in alcohol content. The earliest mention of whiskey in Ireland comes from the Annals of Clonmacnoise, which attributes the death of a chieftain in 1405 to &quot;taking a surfeit of aqua vitae&quot; at Christmas. In Scotland, the first evidence of whisky production comes from an entry in the Exchequer Rolls for 1495 where malt is sent &quot;To Friar John Cor, by order of the king, to make aquavitae&quot;, enough to make about 500 bottles.
==Types==
[[File:Millstone Dutch Single Malt Whisky.jpg|thumb|malt whisky]]
[[File:Haig Club Single Grain Scotch Whisky.jpg|thumb|grain whisky]]
You can split whisky into 2 majors.
*Malt whisky – made from malted barley
*Grain whisky – made from any type of grain
If we go deeper into it, there is nine types of whiskies.
*Rye Whiskey
*Canadian Whiskey 
*Japanese Whiskey
*Bourbon Whiskey
*Tennessee Whiskey 
*Irish Whiskey 
*Scotch Whiskey
*Blended Whiskey
*Single Malt Whiskey
==Chemistry==
Whisky is complex beverage that contains a vast range of flavoring compounds. The flavoring chemicals include carbonyl compounds, alcohols, carboxylic acids and their esters, nitrogen and sulfur-containing compounds, tannins, and other polyphenolic compounds, terpenes, and oxygen-containing, heterocyclic compounds and esters of fatty acids. The nitrogen compounds include pyridines, picolines and pyrazines.

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The phrase &quot;every ism creates a [[w:Schism|schism]]&quot; suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as &quot;isms&quot;) tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. &lt;/ref&gt;&lt;sup&gt;,&lt;/sup&gt;&lt;ref&gt;“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl&lt;/ref&gt; While &quot;isms&quot; can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social &quot;isms&quot; and how they have created [[w:Schism|schisms]] throughout history.

== Political Isms: Capitalism and Communism ==
One of the clearest examples of an &quot;ism&quot; that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership.

This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic &quot;isms&quot; had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations.

== Religious Isms: Protestantism and Catholicism ==
In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an &quot;ism&quot; can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation.

[[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an &quot;ism&quot; grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious &quot;ism&quot; left a legacy of division that altered European history and shaped global religious dynamics.

== Social Isms: Feminism and Patriarchy ==
'''[[w:Feminism|Feminism]]''', another significant &quot;ism,&quot; arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations.

However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism.

Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social &quot;isms&quot; can divide societies.

== Philosophical Isms: Rationalism and Empiricism ==
In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience.

This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual &quot;isms&quot; can divide schools of thought and shape the trajectory of entire fields of inquiry.

== Cultural Isms: Nationalism and Globalism ==
'''[[w:Nationalism|Nationalism]]''' is another &quot;ism&quot; that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism.

The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness.

== Remedies ==
We can gain the wisdom to avoid the schisms born of isms in several ways.

Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful.

It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]].

[[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]].

Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]].

== Conclusion ==
The phrase &quot;every ism creates a schism&quot; captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While &quot;isms&quot; can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas.

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The phrase &quot;every ism creates a [[w:Schism|schism]]&quot; suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as &quot;isms&quot;) tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. &lt;/ref&gt;&lt;sup&gt;,&lt;/sup&gt;&lt;ref&gt;“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl&lt;/ref&gt; While &quot;isms&quot; can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social &quot;isms&quot; and how they have created [[w:Schism|schisms]] throughout history.

== Political Isms: Capitalism and Communism ==
[[File:Berlinermauer.jpg|thumb|The dual “isms” of capitalism and communism created a worldwide “schism” during the Cold War, which manifested physically in the form of heavily fortified borders like the Berlin Wall (fig. 1)]]
One of the clearest examples of an &quot;ism&quot; that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership.

This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic &quot;isms&quot; had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations.

== Religious Isms: Protestantism and Catholicism ==
In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an &quot;ism&quot; can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation.

[[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an &quot;ism&quot; grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious &quot;ism&quot; left a legacy of division that altered European history and shaped global religious dynamics.

== Social Isms: Feminism and Patriarchy ==
'''[[w:Feminism|Feminism]]''', another significant &quot;ism,&quot; arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations.

However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism.

Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social &quot;isms&quot; can divide societies.

== Philosophical Isms: Rationalism and Empiricism ==
In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience.

This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual &quot;isms&quot; can divide schools of thought and shape the trajectory of entire fields of inquiry.

== Cultural Isms: Nationalism and Globalism ==
'''[[w:Nationalism|Nationalism]]''' is another &quot;ism&quot; that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism.

The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness.

== Remedies ==
We can gain the wisdom to avoid the schisms born of isms in several ways.

Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful.

It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]].

[[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]].

Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]].

== Conclusion ==
The phrase &quot;every ism creates a schism&quot; captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While &quot;isms&quot; can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas.

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[[Category:Living Wisely]]</text>
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{{AI-generated}}
The phrase &quot;every ism creates a [[w:Schism|schism]]&quot; suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as &quot;isms&quot;) tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. &lt;/ref&gt;&lt;sup&gt;,&lt;/sup&gt;&lt;ref&gt;“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl&lt;/ref&gt; While &quot;isms&quot; can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social &quot;isms&quot; and how they have created [[w:Schism|schisms]] throughout history.

== Political Isms: Capitalism and Communism ==
[[File:Berlinermauer.jpg|thumb|The dual “isms” of capitalism and communism created a worldwide “schism” during the Cold War, which manifested physically in the form of heavily fortified borders like the Berlin Wall (fig. 1)]]
One of the clearest examples of an &quot;ism&quot; that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership.

This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic &quot;isms&quot; had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations.

== Religious Isms: Protestantism and Catholicism ==
[[File:Peace Line, Belfast - geograph - 1254138.jpg|thumb|A “[[w:Peace lines|peace line]]” in Belfast, another physical manifestation of a “schism” created by “isms” (Protestantism and [[w:Irish republicanism|republicanism]] vs. Catholicism and [[w:Ulster loyalism|loyalism]])]]
In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an &quot;ism&quot; can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation.

[[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an &quot;ism&quot; grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious &quot;ism&quot; left a legacy of division that altered European history and shaped global religious dynamics.

== Social Isms: Feminism and Patriarchy ==
'''[[w:Feminism|Feminism]]''', another significant &quot;ism,&quot; arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations.

However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism.

Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social &quot;isms&quot; can divide societies.

== Philosophical Isms: Rationalism and Empiricism ==
In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience.

This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual &quot;isms&quot; can divide schools of thought and shape the trajectory of entire fields of inquiry.

== Cultural Isms: Nationalism and Globalism ==
'''[[w:Nationalism|Nationalism]]''' is another &quot;ism&quot; that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism.

The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness.

== Remedies ==
We can gain the wisdom to avoid the schisms born of isms in several ways.

Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful.

It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]].

[[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]].

Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]].

== Conclusion ==
The phrase &quot;every ism creates a schism&quot; captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While &quot;isms&quot; can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas.

{{CourseCat}}
[[Category:Essays]]
[[Category:Living Wisely]]</text>
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{{AI-generated}}
The phrase &quot;every ism creates a [[w:Schism|schism]]&quot; suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as &quot;isms&quot;) tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. &lt;/ref&gt;&lt;sup&gt;,&lt;/sup&gt;&lt;ref&gt;“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl&lt;/ref&gt; While &quot;isms&quot; can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social &quot;isms&quot; and how they have created [[w:Schism|schisms]] throughout history.

== Political Isms: Capitalism and Communism ==
[[File:Berlinermauer.jpg|thumb|The dual “isms” of capitalism and communism created a worldwide “schism” during the Cold War, which manifested physically in the form of heavily fortified borders like the Berlin Wall (fig. 1)]]
One of the clearest examples of an &quot;ism&quot; that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership.

This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic &quot;isms&quot; had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations.

== Religious Isms: Protestantism and Catholicism ==
[[File:Peace Line, Belfast - geograph - 1254138.jpg|thumb|A “[[w:Peace lines|peace line]]” in Belfast, another physical manifestation of a “schism” created by “isms” (Catholicism and [[w:Irish republicanism|republicanism]] vs. Protestantism  and [[w: Unionism in Ireland|unionism]])]]
In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an &quot;ism&quot; can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation.

[[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an &quot;ism&quot; grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious &quot;ism&quot; left a legacy of division that altered European history and shaped global religious dynamics.

== Social Isms: Feminism and Patriarchy ==
'''[[w:Feminism|Feminism]]''', another significant &quot;ism,&quot; arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations.

However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism.

Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social &quot;isms&quot; can divide societies.

== Philosophical Isms: Rationalism and Empiricism ==
In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience.

This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual &quot;isms&quot; can divide schools of thought and shape the trajectory of entire fields of inquiry.

== Cultural Isms: Nationalism and Globalism ==
'''[[w:Nationalism|Nationalism]]''' is another &quot;ism&quot; that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism.

The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness.

== Remedies ==
We can gain the wisdom to avoid the schisms born of isms in several ways.

Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful.

It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]].

[[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]].

Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]].

== Conclusion ==
The phrase &quot;every ism creates a schism&quot; captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While &quot;isms&quot; can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas.

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[[Category:Living Wisely]]</text>
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{{AI-generated}}
The phrase &quot;every ism creates a [[w:Schism|schism]]&quot; suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as &quot;isms&quot;) tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. &lt;/ref&gt;&lt;sup&gt;,&lt;/sup&gt;&lt;ref&gt;“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl&lt;/ref&gt; While &quot;isms&quot; can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social &quot;isms&quot; and how they have created [[w:Schism|schisms]] throughout history.

== Political Isms: Capitalism and Communism ==
[[File:Berlinermauer.jpg|thumb|The dual “isms” of capitalism and communism created a worldwide “schism” during the Cold War, which manifested physically in the form of heavily fortified borders like the Berlin Wall (fig. 1)]]
One of the clearest examples of an &quot;ism&quot; that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership.

This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic &quot;isms&quot; had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations.

== Religious Isms: Protestantism and Catholicism ==
[[File:Peace Line, Belfast - geograph - 1254138.jpg|thumb|A “[[w:Peace lines|peace line]]” in Belfast, another physical manifestation of a “schism” ([[w:The Troubles|the Troubles]]) created by “isms” (Catholicism and [[w:Irish republicanism|republicanism]] vs. Protestantism  and [[w: Unionism in Ireland|unionism]])]]
In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an &quot;ism&quot; can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation.

[[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an &quot;ism&quot; grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious &quot;ism&quot; left a legacy of division that altered European history and shaped global religious dynamics.

== Social Isms: Feminism and Patriarchy ==
'''[[w:Feminism|Feminism]]''', another significant &quot;ism,&quot; arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations.

However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism.

Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social &quot;isms&quot; can divide societies.

== Philosophical Isms: Rationalism and Empiricism ==
In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience.

This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual &quot;isms&quot; can divide schools of thought and shape the trajectory of entire fields of inquiry.

== Cultural Isms: Nationalism and Globalism ==
'''[[w:Nationalism|Nationalism]]''' is another &quot;ism&quot; that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism.

The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness.

== Remedies ==
We can gain the wisdom to avoid the schisms born of isms in several ways.

Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful.

It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]].

[[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]].

Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]].

== Conclusion ==
The phrase &quot;every ism creates a schism&quot; captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While &quot;isms&quot; can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas.

{{CourseCat}}
[[Category:Essays]]
[[Category:Living Wisely]]</text>
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{{AI-generated}}
The phrase &quot;every ism creates a [[w:Schism|schism]]&quot; suggests that [[w:Ideology|ideologies]], philosophies, and belief systems (referred to as &quot;isms&quot;) tend to divide people into opposing factions or camps, often leading to conflict, misunderstanding, or alienation.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] generated the first draft of this text responding to the prompt: “Write an essay exploring the phrase ‘every ism creates a schism.. Provide examples”. It has been edited subsequently. &lt;/ref&gt;&lt;sup&gt;,&lt;/sup&gt;&lt;ref&gt;“[https://medium.com/age-of-awareness/every-ism-creates-a-schism-avoiding-the-habit-of-othering-283594a5eb8c Every -ism creates a schism]”- Avoiding the habit of othering, Jan 28, 2020, Daniel Christian Wahl&lt;/ref&gt; While &quot;isms&quot; can be powerful forces for social, political, and intellectual change, they frequently introduce division by rigidly categorizing beliefs and identities, pushing people to define themselves as either for or against a particular stance. This essay will explore this idea by examining historical, political, religious, and social &quot;isms&quot; and how they have created [[w:Schism|schisms]] throughout history.

== Political Isms: Capitalism and Communism ==
[[File:Berlinermauer.jpg|thumb|The dual “isms” of capitalism and communism created a worldwide “schism” during the Cold War, which manifested physically in the form of heavily fortified borders like the Berlin Wall]]
One of the clearest examples of an &quot;ism&quot; that has created a profound schism is the divide between '''[[w:Capitalism|capitalism]]''' and '''[[w:Communism|communism]]''' in the 20th century. These two economic ideologies, based on fundamentally different views of ownership, wealth distribution, and the role of the state, polarized much of the world during the [[w:Cold_War|Cold War]] era. Capitalism, with its emphasis on [[w:Free_market|free markets]] and private property, contrasted sharply with communism's ideals of state control and communal ownership.

This ideological divide led to the formation of opposing power blocs: the '''[[w:Western_Bloc|Western capitalist countries]]''', led by the United States, and the '''[[w:Eastern_Bloc|Eastern communist bloc]]''', led by the Soviet Union. The schism was not just theoretical—it fueled political, economic, and military conflicts, such as the '''[[w:Korean_War|Korean War]]''', '''[[w:Vietnam_War|Vietnam War]]''', and various proxy battles around the globe. The schism created by these economic &quot;isms&quot; had devastating effects, entrenching divisions that still linger in geopolitics today, as seen in ongoing tensions between capitalist and communist or post-communist nations.

== Religious Isms: Protestantism and Catholicism ==
[[File:Peace Line, Belfast - geograph - 1254138.jpg|thumb|A “[[w:Peace lines|peace line]]” in Belfast, another physical manifestation of a “schism” ([[w:The Troubles|the Troubles]]) created by “isms” (Catholicism and [[w:Irish republicanism|republicanism]] vs. Protestantism  and [[w: Unionism in Ireland|unionism]])]]
In the realm of religion, the [[w:Reformation|Reformation]] in the 16th century is a prime example of how an &quot;ism&quot; can create a lasting schism. The emergence of '''[[w:Protestantism|Protestantism]]''' as a reform movement against certain practices of the '''[[w:Catholic_Church|Catholic Church]]''' led to a division that not only altered the religious landscape of Europe but also caused political upheavals, wars, and social fragmentation.

[[w:Martin_Luther|Martin Luther’s]] critique of the Catholic Church’s practices, such as the selling of [[w:Indulgence|indulgences]], gave birth to Protestantism, an &quot;ism&quot; grounded in the belief of personal faith over institutionalized authority. This led to a profound schism, splitting Christianity into two major branches. The divide sparked religious wars like the '''[[w:Thirty_Years'_War|Thirty Years' War]]''', which devastated much of Europe, and continues to influence tensions between Protestant and Catholic communities, particularly in regions like [[w:Northern_Ireland|Northern Ireland]]. The schism brought about by this religious &quot;ism&quot; left a legacy of division that altered European history and shaped global religious dynamics.

== Social Isms: Feminism and Patriarchy ==
'''[[w:Feminism|Feminism]]''', another significant &quot;ism,&quot; arose in response to the historical domination of '''[[w:Patriarchy|patriarchy]]''', the social system in which men hold power and dominate in roles of leadership, moral authority, and social privilege. Feminism, especially since the 19th century, has fought for the rights of women to vote, work, and live free of oppression, fundamentally challenging patriarchal norms and expectations.

However, feminism has created its own internal schisms. The early feminist movement often focused on the concerns of middle-class white women, leading to a divide between '''[[w:White_feminism|white feminism]]''' and '''[[w:Intersectionality|intersectional]] feminism''', the latter of which emphasizes the overlapping and interconnected forms of oppression that include race, class, and sexuality. For example, the divide between the concerns of black feminists and the mainstream feminist movement became more pronounced during the '''[[w:Civil_rights_movement|civil rights era]]''', highlighting how even within a movement, different experiences of oppression can lead to schism.

Additionally, feminism has created tension between those who resist change and those who advocate for [[w:Gender_equality|gender equality]]. Opponents of feminism often see it as a threat to traditional values, leading to cultural and political battles over issues like [[w:Reproductive_rights|reproductive rights]], workplace equality, and the gender pay gap. This ongoing schism shows how deeply entrenched social &quot;isms&quot; can divide societies.

== Philosophical Isms: Rationalism and Empiricism ==
In [[philosophy]], the schism between '''[[w:Rationalism|rationalism]]''' and '''[[w:Empiricism|empiricism]]''' has shaped much of Western thought. Rationalism, championed by figures like '''[[w:René_Descartes|René Descartes]]''', argues that knowledge is primarily acquired through reason and logical deduction. In contrast, empiricism, advocated by thinkers like '''[[w:John_Locke|John Locke]]''' and '''[[w:David_Hume|David Hume]]''', posits that knowledge comes primarily from sensory experience.

This philosophical schism has led to deep debates within [[Knowing How You Know|epistemology]], the branch of philosophy concerned with the nature of knowledge. Rationalists and empiricists offer opposing views on how we come to know and understand the world, with implications for science, ethics, and [[w:Metaphysics|metaphysics]]. The rationalist-empiricist schism exemplifies how intellectual &quot;isms&quot; can divide schools of thought and shape the trajectory of entire fields of inquiry.

== Cultural Isms: Nationalism and Globalism ==
'''[[w:Nationalism|Nationalism]]''' is another &quot;ism&quot; that has often led to schism. Defined as a strong identification with and loyalty to one's nation, nationalism has been a driving force behind the formation of nation-states, independence movements, and wars. The rise of '''[[w:Globalism|globalism]]''', the idea that nations and cultures are interconnected and that global cooperation is essential for addressing shared challenges, presents a direct challenge to nationalism.

The schism between nationalism and globalism is evident in modern political debates. Nationalist movements often prioritize sovereignty, border control, and economic self-sufficiency, while globalists emphasize international trade, environmental cooperation, and multiculturalism. This divide has become especially apparent in debates over issues like immigration, climate change, and trade agreements. Events such as '''[[w:Brexit|Brexit]]''' and the rise of [[w:Populism|populist]] leaders in various countries underscore the schism between those who favor nationalism and those who advocate for global interconnectedness.

== Remedies ==
We can gain the wisdom to avoid the schisms born of isms in several ways.

Begin by [[Facing Facts#Degrees of Consensus|separating facts from fiction]], speculation, opinions, and controversies. [[Knowing How You Know|Know how you know]] and [[Seeking True Beliefs|seek true beliefs]]. [[Knowing How You Know/Examining Ideologies|Examine the various ideologies]] you are drawn to. Abandon those that are unsound or unhelpful.

It is also helpful to recognize that because [[Embracing Ambiguity/Ambiguity breeds schisms|ambiguity breeds schisms]] it is helpful to [[Embracing Ambiguity|embrace ambiguity]], [[Practicing Dialogue|practice dialogue]], and [[Transcending Conflict|transcend conflict]].

[[Living Wisely/Seeking Real Good|Seek real good]] and [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|transcend ideology]].

Work to [[Finding Common Ground|find common ground]] and [[Coming Together|come together]].

== Conclusion ==
The phrase &quot;every ism creates a schism&quot; captures a profound truth about human societies: the creation of any organized belief system, whether political, religious, social, or philosophical, often introduces division. While &quot;isms&quot; can provide clarity, identity, and a sense of belonging, they also have the potential to alienate and divide, leading to ideological rifts and conflicts. As we have seen through the examples of capitalism vs. communism, Protestantism vs. Catholicism, feminism vs. patriarchy, rationalism vs. empiricism, and nationalism vs. globalism, these divisions shape not only intellectual debates but also the course of history. Understanding these schisms helps us navigate the complexities of belief and coexistence in a world full of competing ideas.

{{CourseCat}}
[[Category:Essays]]
[[Category:Living Wisely]]</text>
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  <page>
    <title>WikiJournal Preprints/24-cell</title>
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        <id>2856930</id>
      </contributor>
      <comment>/* Hexagons and &lt;s&gt;hexagrams&lt;/s&gt; */ dodecagram should replace hexagram everywhere, since we discovered that the Clifford polygon of the hexagonal rotation is a {12/5} dodecagram, not a {6/2} hexagram as we previously stated -- this update is in progress in [[24-cell]] but has not been applied to this preprint version of the article yet</comment>
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|journal=Wikijournal Preprints
|last=Christie
|first=David Brooks
|abstract=The 24-cell is one of only a few uniform polytopes in which the edge length equals the radius. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the geometry of the 24-cell in detail, as an expression of its rotational symmetries.
|w1=24-cell
}}

== The unique 24-point 24-cell polytope ==

The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.

== The 24-cell in the proper sequence of 4-polytopes ==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest (and sharpest) case, and the 120-cell is the largest (and roundest). Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope.

The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

== Coordinates ==
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

=== Great squares ===
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]].

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

=== Great hexagons ===
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the [[#Great squares|great squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

=== Great triangles ===
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} 

Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.

== Hypercubic chords ==
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

== Geodesics ==
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

== Constructions ==
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue),{{Sfn|Egan|2019|ps=; Double-rotating 24-cell with orthogonal red, green and blue vertices.}} double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

== Relationships among interior polytopes ==
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

== Boundary cells ==
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

== Radially equilateral honeycomb ==
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;&lt;s&gt;{{Efn|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] &lt;s&gt;[[W:Hexagram|hexagram]]&lt;/s&gt; with {{radic|3}} edges.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic]]&lt;/s&gt; to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted &lt;s&gt;{6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt;.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of &lt;s&gt;6 vertices (hexagrams)&lt;/s&gt; that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|&lt;s&gt;hexagram&lt;/s&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their &lt;s&gt;[[#Isoclinic rotations|isoclinic helix hexagrams]]&lt;/s&gt;.{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s&lt;s&gt;{{Efn|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}}

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical &lt;s&gt;hexagrams&lt;/s&gt; and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical &lt;s&gt;hexagram&lt;/s&gt; isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew &lt;s&gt;[[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The &lt;s&gt;hexagram&lt;/s&gt; does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] &lt;s&gt;hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop&lt;/s&gt;.{{Efn|name=double threaded}}

Each 6-cell ring contains &lt;s&gt;six such hexagram&lt;/s&gt; isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic &lt;s&gt;hexagram&lt;/s&gt; geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew &lt;s&gt;[[W:Hexagram|hexagram]]s&lt;/s&gt; lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white &lt;s&gt;hexagrams&lt;/s&gt; pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew &lt;s&gt;hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram&lt;sub&gt;2&lt;/sub&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|-
|colspan=5|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|-
|colspan=15|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}

== Conclusions ==
Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}

== Acknowledgements ==
This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for three indispensible rotating animations, one created by Greg Egan and two by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise (Jason Hise)]], which I have retained with attribution. Those images and others which appear in my tables and footnotes{{Efn|I am the author of the footnotes to this article, except for quotations and images they contain.}} are from Wikimedia Commons, with attributions; most were created by Wikipedia editor and illustrator [[W:User:Tomruen|Tomruen (Tom Ruen)]]. Consequently, this version is not a complete treatment of the 24-cell; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]].

Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other source treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood most readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into a single encyclopedic hypertext. Well-illustrated hypertext seems naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations.

== Future work ==
This paper is part of the evolving [[Polyscheme#Polyscheme project articles|Polyscheme collection of articles]] hosted at Wikiversity by the [[Polyscheme]] learning project.

The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for an expanded, more-than-encyclopedic version of it and the other 4-polytope articles I was engaged in editing, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles.

Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students, educators and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles.

Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell#24-cell|24-cell article hosted at Wikiversity]] as part of the [[Polyscheme|Polyscheme research project]], which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should rightly be an abridged version of that expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been excluded.

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}


* {{Citation|author-last=Hise|author-first=Jason|date=2011|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a simple rotation|title-link=Wikimedia:File:24-cell.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Hise|author-first=Jason|date=2007|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a double rotation|title-link=Wikimedia:File:24-cell-orig.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Egan|author-first=Greg|date=2019|title=A 24-cell containing red, green, and blue 16-cells performing a double rotation|title-link=Wikimedia:File:24-cell-3CP.gif|journal=Wikimedia Commons}}
{{Refend}}</text>
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|journal=Wikijournal Preprints
|last=Christie
|first=David Brooks
|abstract=The 24-cell is one of only a few uniform polytopes in which the edge length equals the radius. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the geometry of the 24-cell in detail, as an expression of its rotational symmetries.
|w1=24-cell
}}

== The unique 24-point 24-cell polytope ==

The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.

== The 24-cell in the proper sequence of 4-polytopes ==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest (and sharpest) case, and the 120-cell is the largest (and roundest). Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope.

The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

== Coordinates ==
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

=== Great squares ===
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]].

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

=== Great hexagons ===
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the [[#Great squares|great squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

=== Great triangles ===
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} 

Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.

== Hypercubic chords ==
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

== Geodesics ==
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

== Constructions ==
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue),{{Sfn|Egan|2019|ps=; Double-rotating 24-cell with orthogonal red, green and blue vertices.}} double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

== Relationships among interior polytopes ==
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

== Boundary cells ==
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

== Radially equilateral honeycomb ==
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;&lt;s&gt;{{Efn|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] &lt;s&gt;[[W:Hexagram|hexagram]]&lt;/s&gt; with {{radic|3}} edges.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic]]&lt;/s&gt; to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted &lt;s&gt;{6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt;.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of &lt;s&gt;6 vertices (hexagrams)&lt;/s&gt; that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|&lt;s&gt;hexagram&lt;/s&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their &lt;s&gt;[[#Isoclinic rotations|isoclinic helix hexagrams]]&lt;/s&gt;.{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s&lt;s&gt;{{Efn|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}}

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical &lt;s&gt;hexagrams&lt;/s&gt; and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical &lt;s&gt;hexagram&lt;/s&gt; isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew &lt;s&gt;[[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The &lt;s&gt;hexagram&lt;/s&gt; does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] &lt;s&gt;hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop&lt;/s&gt;.{{Efn|name=double threaded}}

Each 6-cell ring contains &lt;s&gt;six such hexagram&lt;/s&gt; isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic &lt;s&gt;hexagram&lt;/s&gt; geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew &lt;s&gt;[[W:Hexagram|hexagram]]s&lt;/s&gt; lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white &lt;s&gt;hexagrams&lt;/s&gt; pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew &lt;s&gt;hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}}&lt;/s&gt; contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.&lt;s&gt;{{Efn|Each vertex of the 6-cell ring is intersected by two skew &lt;s&gt;hexagrams&lt;/s&gt; of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams&lt;/s&gt; hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew &lt;s&gt;hexagrams&lt;/s&gt;, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}}&lt;/s&gt; Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.&lt;s&gt;{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,&lt;s&gt;{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop &lt;s&gt;hexagram,{{Efn|name=Möbius double loop hexagram}}&lt;/s&gt; which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}}&lt;/s&gt; missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.&lt;s&gt;{{Efn|name=Möbius double loop hexagram}}&lt;/s&gt;|name=isoclines at hexagons}} and four &lt;s&gt;hexagram&lt;/s&gt; isoclines (all black or all white) that cross at the vertex.&lt;s&gt;{{Efn|Each &lt;s&gt;hexagram&lt;/s&gt; isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}}&lt;/s&gt; Four &lt;s&gt;hexagram&lt;/s&gt; isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct &lt;s&gt;hexagram&lt;/s&gt; isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of &lt;s&gt;hexagram&lt;/s&gt; isoclines in each fibration&lt;s&gt;{{Efn|name=hexagram isoclines at an axis}}&lt;/s&gt; and the 16 distinct &lt;s&gt;hexagram&lt;/s&gt; isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 &lt;s&gt;hexagram&lt;/s&gt; isoclines, and each cell ring contains 3 black-white pairs of the 16 &lt;s&gt;hexagram&lt;/s&gt; isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|-
|colspan=5|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]]&lt;/s&gt; that winds twice around the 3-sphere on every ''second'' vertex of the &lt;s&gt;hexagram&lt;/s&gt;. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} &lt;s&gt;[[W:Hexagram|hexagram]]&lt;/s&gt; with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The &lt;s&gt;hexagram&lt;/s&gt; projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram isoclines]]&lt;/s&gt; are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew &lt;s&gt;hexagram&lt;/s&gt; isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford &lt;s&gt;hexagram&lt;/s&gt;, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 &lt;s&gt;hexagram&lt;/s&gt; isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|-
|colspan=15|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew &lt;s&gt;[[W:Hexagram|hexagram]]s&lt;/s&gt;, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}

== Conclusions ==
Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}

== Acknowledgements ==
This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for three indispensible rotating animations, one created by Greg Egan and two by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise (Jason Hise)]], which I have retained with attribution. Those images and others which appear in my tables and footnotes{{Efn|I am the author of the footnotes to this article, except for quotations and images they contain.}} are from Wikimedia Commons, with attributions; most were created by Wikipedia editor and illustrator [[W:User:Tomruen|Tomruen (Tom Ruen)]]. Consequently, this version is not a complete treatment of the 24-cell; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]].

Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other source treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood most readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into a single encyclopedic hypertext. Well-illustrated hypertext seems naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations.

== Future work ==
This paper is part of the evolving [[Polyscheme#Polyscheme project articles|Polyscheme collection of articles]] hosted at Wikiversity by the [[Polyscheme]] learning project.

The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for an expanded, more-than-encyclopedic version of it and the other 4-polytope articles I was engaged in editing, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles.

Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students, educators and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles.

Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell#24-cell|24-cell article hosted at Wikiversity]] as part of the [[Polyscheme|Polyscheme research project]], which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should rightly be an abridged version of that expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been excluded.

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}


* {{Citation|author-last=Hise|author-first=Jason|date=2011|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a simple rotation|title-link=Wikimedia:File:24-cell.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Hise|author-first=Jason|date=2007|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a double rotation|title-link=Wikimedia:File:24-cell-orig.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Egan|author-first=Greg|date=2019|title=A 24-cell containing red, green, and blue 16-cells performing a double rotation|title-link=Wikimedia:File:24-cell-3CP.gif|journal=Wikimedia Commons}}
{{Refend}}</text>
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|journal=Wikijournal Preprints
|last=Christie
|first=David Brooks
|abstract=The 24-cell is one of only a few uniform polytopes in which the edge length equals the radius. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the geometry of the 24-cell in detail, as an expression of its rotational symmetries.
|w1=24-cell
}}

== The unique 24-point 24-cell polytope ==

The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.

== The 24-cell in the proper sequence of 4-polytopes ==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest (and sharpest) case, and the 120-cell is the largest (and roundest). Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope.

The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}}

== Coordinates ==
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

=== Great squares ===
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]].

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

=== Great hexagons ===
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the [[#Great squares|great squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

=== Great triangles ===
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} 

Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.

== Hypercubic chords ==
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

== Geodesics ==
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

== Constructions ==
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue),{{Sfn|Egan|2019|ps=; Double-rotating 24-cell with orthogonal red, green and blue vertices.}} double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

== Relationships among interior polytopes ==
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

== Boundary cells ==
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

== Radially equilateral honeycomb ==
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;&lt;s&gt;{{Efn|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] &lt;s&gt;[[W:Hexagram|hexagram]]&lt;/s&gt; with {{radic|3}} edges.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic]]&lt;/s&gt; to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted &lt;s&gt;{6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt;.&lt;s&gt;{{Efn|name=skew hexagram}}&lt;/s&gt; Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of &lt;s&gt;6 vertices (hexagrams)&lt;/s&gt; that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|&lt;s&gt;hexagram&lt;/s&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the &quot;left&quot; is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their &lt;s&gt;[[#Isoclinic rotations|isoclinic helix hexagrams]]&lt;/s&gt;.{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s&lt;s&gt;{{Efn|name=Petrie dodecagram and Clifford hexagram}}&lt;/s&gt; and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}}

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical &lt;s&gt;hexagrams&lt;/s&gt; and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical &lt;s&gt;hexagram&lt;/s&gt; isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew &lt;s&gt;[[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The &lt;s&gt;hexagram&lt;/s&gt; does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] &lt;s&gt;hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop&lt;/s&gt;.{{Efn|name=double threaded}}

Each 6-cell ring contains &lt;s&gt;six such hexagram&lt;/s&gt; isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic &lt;s&gt;hexagram&lt;/s&gt; geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew &lt;s&gt;[[W:Hexagram|hexagram]]s&lt;/s&gt; lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white &lt;s&gt;hexagrams&lt;/s&gt; pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew &lt;s&gt;hexagram&lt;/s&gt;{{Efn|Each half of a skew &lt;s&gt;hexagram&lt;/s&gt; is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew &lt;s&gt;hexagrams&lt;/s&gt; of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew &lt;s&gt;hexagrams&lt;/s&gt;, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic &lt;s&gt;hexagram&lt;sub&gt;2&lt;/sub&gt;&lt;/s&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew &lt;s&gt;hexagram&lt;/s&gt; and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop &lt;s&gt;hexagram,{{Efn|name=Möbius double loop hexagram}}&lt;/s&gt; which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.&lt;s&gt;{{Efn|name=Möbius double loop hexagram}}&lt;/s&gt;|name=isoclines at hexagons}} and four &lt;s&gt;hexagram&lt;/s&gt; isoclines (all black or all white) that cross at the vertex.{{Efn|Each &lt;s&gt;hexagram&lt;/s&gt; isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four &lt;s&gt;hexagram&lt;/s&gt; isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct &lt;s&gt;hexagram&lt;/s&gt; isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of &lt;s&gt;hexagram&lt;/s&gt; isoclines in each fibration&lt;s&gt;{{Efn|name=hexagram isoclines at an axis}}&lt;/s&gt; and the 16 distinct &lt;s&gt;hexagram&lt;/s&gt; isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 &lt;s&gt;hexagram&lt;/s&gt; isoclines, and each cell ring contains 3 black-white pairs of the 16 &lt;s&gt;hexagram&lt;/s&gt; isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|-
|colspan=5|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
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|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]]&lt;/s&gt; that winds twice around the 3-sphere on every ''second'' vertex of the &lt;s&gt;hexagram&lt;/s&gt;. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|&lt;blockquote&gt;Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions).&lt;br&gt;&lt;br&gt;Every orthogonal transformation is expressible as{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;&lt;br&gt;For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}&lt;/blockquote&gt;|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']]{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} &lt;s&gt;[[W:Hexagram|hexagram]]&lt;/s&gt; with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The &lt;s&gt;hexagram&lt;/s&gt; projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew &lt;s&gt;[[#Helical hexagrams and their isoclines|hexagram isoclines]]&lt;/s&gt; are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew &lt;s&gt;hexagram&lt;/s&gt; isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford &lt;s&gt;hexagram&lt;/s&gt;, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 &lt;s&gt;hexagram&lt;/s&gt; isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;{{text color default}};&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;{{text color default}};&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;{{text color default}};&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|-
|colspan=15|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}}
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew &lt;s&gt;[[W:Hexagram|hexagram]]s&lt;/s&gt;, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}

== Conclusions ==
Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}

== Acknowledgements ==
This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for three indispensible rotating animations, one created by Greg Egan and two by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise (Jason Hise)]], which I have retained with attribution. Those images and others which appear in my tables and footnotes{{Efn|I am the author of the footnotes to this article, except for quotations and images they contain.}} are from Wikimedia Commons, with attributions; most were created by Wikipedia editor and illustrator [[W:User:Tomruen|Tomruen (Tom Ruen)]]. Consequently, this version is not a complete treatment of the 24-cell; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]].

Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other source treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood most readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into a single encyclopedic hypertext. Well-illustrated hypertext seems naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations.

== Future work ==
This paper is part of the evolving [[Polyscheme#Polyscheme project articles|Polyscheme collection of articles]] hosted at Wikiversity by the [[Polyscheme]] learning project.

The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for an expanded, more-than-encyclopedic version of it and the other 4-polytope articles I was engaged in editing, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles.

Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students, educators and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles.

Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell#24-cell|24-cell article hosted at Wikiversity]] as part of the [[Polyscheme|Polyscheme research project]], which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should rightly be an abridged version of that expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been excluded.

== Notes ==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

== Citations ==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

== References ==
{{Refbegin}}
{{Regular convex 4-polytopes Refs|wiki=W:}}


* {{Citation|author-last=Hise|author-first=Jason|date=2011|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a simple rotation|title-link=Wikimedia:File:24-cell.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Hise|author-first=Jason|date=2007|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a double rotation|title-link=Wikimedia:File:24-cell-orig.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Egan|author-first=Greg|date=2019|title=A 24-cell containing red, green, and blue 16-cells performing a double rotation|title-link=Wikimedia:File:24-cell-3CP.gif|journal=Wikimedia Commons}}
{{Refend}}</text>
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[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (May 21, 2026), I've made just over 393,000 edits to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]].

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
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    <title>Einstein Probability Dilation</title>
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{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

= Scope and limitations =
EPD is a mathematical framework for measure transformations. It does not claim:
* a replacement theory for GR/QM,
* empirical confirmation without explicit predictions and tests.


The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

Recent mathematical work extending Buffon's needle problem to curved manifolds suggests a possible connection between probability distributions and intrinsic geometry. In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.&lt;ref&gt;Feynman, R. P., &amp; Hibbs, A. R. (1965). ''Quantum Mechanics and Path Integrals''. McGraw-Hill.&lt;/ref&gt;

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where:

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

represents the standard quantum probability amplitude,

&lt;math&gt;W(K)&lt;/math&gt;

represents a curvature-dependent geometric weighting factor,

and

&lt;math&gt;K&lt;/math&gt;

represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:
* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where:

&lt;math&gt;L&lt;/math&gt;

represents a microscopic path scale,

&lt;math&gt;K&lt;/math&gt;

represents local curvature,

and

&lt;math&gt;c&lt;/math&gt;

represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:
* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.
==== Open Questions ====

* Can curvature-weighted path measures be formally derived?
* Can Buffon deficit relations be generalized to relativistic spacetime?
* Could curvature-weighting regulate ultraviolet divergences?
* What relationship exists between Gaussian curvature and spacetime curvature tensors?
* Could probabilistic geometry contribute to emergent spacetime models?

= Future directions =
* develop canonical families of dilation fields and invariants,
* clarify “structure-from-measure” diagnostics,
* publish reproducible simulation notebooks and parameter sweeps,
* compare multiple dilation families under shared evaluation criteria.

= See also =
* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

== Status of the Framework ==
Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model. Several portions of the article discuss heuristic interpretations and proposed extensions which are not established physical theory.

The article is intended as an educational and exploratory investigation into possible relationships between probability structure, geometric curvature, and quantum path behavior.

= References =
==== Probability / measure theory ====
* [[w:Measure (mathematics)|Measure (mathematics)]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]&lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19&lt;/ref&gt;
* &lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19&lt;/ref&gt;

==== Computing / simulation ====
* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==
© Howard Richardson. Licensed under CC BY 4.0 International.

Reuse permitted with attribution.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.

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{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.

{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.
== Scope and limitations ==
EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics,
* empirical confirmation without explicit predictions and tests.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==
{{Disclaimer|This section is speculative and exploratory in nature.}}

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19&lt;/ref&gt;

In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.&lt;ref&gt;Feynman, R. P., &amp; Hibbs, A. R. (1965). ''Quantum Mechanics and Path Integrals''. McGraw-Hill.&lt;/ref&gt;

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where:

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

represents the standard quantum probability amplitude,

&lt;math&gt;W(K)&lt;/math&gt;

represents a curvature-dependent geometric weighting factor,

and

&lt;math&gt;K&lt;/math&gt;

represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:

* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where:

&lt;math&gt;L&lt;/math&gt;

represents a microscopic path scale,

&lt;math&gt;K&lt;/math&gt;

represents local curvature,

and

&lt;math&gt;c&lt;/math&gt;

represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:

* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived?
* Can Buffon deficit relations be generalized to relativistic spacetime?
* Could curvature-weighting regulate ultraviolet divergences?
* What relationship exists between Gaussian curvature and spacetime curvature tensors?
* Could probabilistic geometry contribute to emergent spacetime models?

== Future directions ==

* develop canonical families of dilation fields and invariants,
* clarify “structure-from-measure” diagnostics,
* publish reproducible simulation notebooks and parameter sweeps,
* compare multiple dilation families under shared evaluation criteria.

== Status of the Framework ==
Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model. Several portions of the article discuss heuristic interpretations and proposed extensions which are not established physical theory.

The article is intended as an educational and exploratory investigation into possible relationships between probability structure, geometric curvature, and quantum path behavior.

== See also ==

* [[wikipedia:Buffon's needle problem|Buffon's needle problem]]
* [[wikipedia:Probability measure|Probability measure]]
* [[wikipedia:Cosmological principle|Cosmological principle]]
* [[wikipedia:Holographic principle|Holographic principle]]
* [[wikipedia:Fractal geometry|Fractal geometry]]
* [[wikipedia:Quantum field theory|Quantum field theory]]
* [[wikipedia:Langlands program|Langlands program]]

== References ==
&lt;references /&gt;

=== Further reading ===

==== Probability and measure theory ====

* [[wikipedia:Measure (mathematics)|Measure]]
* [[wikipedia:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[wikipedia:Probability theory|Probability theory]]

==== Computing and simulation ====

* [[wikipedia:Monte Carlo method|Monte Carlo method]]
* [[wikipedia:Importance sampling|Importance sampling]]

== Copyright and licensing ==
© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
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      <text bytes="43744" sha1="qyf6pemvk276obk76yhu9qbbvvggu3l" xml:space="preserve">{{Research project}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be a finite set, such as bins, configurations, or catalog points. The probability measure &lt;math&gt;P&lt;/math&gt; may then be represented as a probability vector or as an empirical measure over those states.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing &lt;math&gt;\xi(r)&lt;/math&gt; ===

This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the monopole two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal is not a precision cosmology fit, but a clean, falsifiable pipeline.

The demonstration proceeds as follows:

* generate a baseline mock galaxy catalog in a periodic box;
* define a strictly positive dilation field over galaxy positions;
* perform EPD-style importance resampling to obtain a reweighted mock catalog;
* measure &lt;math&gt;\xi(r)&lt;/math&gt; using the normalized Landy–Szalay estimator;
* tune a small parameter grid to minimize a simple mismatch score on selected distance bins.

The dilation field may be written as:

&lt;math&gt;
D(x)=\exp(\lambda \phi(x))
&lt;/math&gt;

where &lt;math&gt;\lambda&gt;0&lt;/math&gt; is a dilation-strength parameter and &lt;math&gt;\phi(x)\geq 0&lt;/math&gt; is a nonnegative field defined over positions.

One example field is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed points and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in the relevant catalogs.

{{Note|Unless observational datasets are explicitly supplied, this demonstration may be run using synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run with a synthetic target curve, the baseline mock catalog should remain close to &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock may show enhanced small-scale clustering relative to the baseline mock catalog. This illustrates the reweighting mechanism but does not constitute observational confirmation.

The full Python demonstration code should preferably be placed on a Wikiversity subpage, GitHub repository, OSF component, or other supplementary location, rather than embedded in full in the main article body.

== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.&lt;ref&gt;
Feynman, R. P., &amp; Hibbs, A. R. (1965). ''Quantum Mechanics and Path Integrals''. McGraw-Hill.
&lt;/ref&gt;

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude, &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent geometric weighting factor, and &lt;math&gt;K&lt;/math&gt; represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;
e^{iS/\hbar}
&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:

* a quantum interference structure;
* a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale, &lt;math&gt;K&lt;/math&gt; represents local curvature, and &lt;math&gt;c&lt;/math&gt; represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow 0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:

* stochastic geometry;
* probabilistic curvature;
* quantum path integrals;
* emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived?
* Can Buffon deficit relations be generalized to relativistic spacetime?
* Could curvature-weighting regulate ultraviolet divergences?
* What relationship exists between Gaussian curvature and spacetime curvature tensors?
* Could probabilistic geometry contribute to emergent spacetime models?

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria.

== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model. Several portions of the article discuss heuristic interpretations and proposed extensions which are not established physical theory.

The article is intended as an educational and exploratory investigation into possible relationships between probability structure, geometric curvature, and quantum path behavior.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Cosmological principle|Cosmological principle]]
* [[w:Holographic principle|Holographic principle]]
* [[w:Fractal geometry|Fractal geometry]]
* [[w:Quantum field theory|Quantum field theory]]
* [[w:Langlands program|Langlands program]]

== References ==

&lt;references/&gt;

=== Further reading ===

==== Probability and measure theory ====

* [[w:Measure (mathematics)|Measure]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Probability theory|Probability theory]]

==== Computing and simulation ====

* [[w:Monte Carlo method|Monte Carlo method]]
* [[w:Importance sampling|Importance sampling]]

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant: &lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure, &lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}\quad\text{for all }A\in\Sigma.&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt; where

&lt;math&gt;\widetilde{p}(x)=\frac{D(x)\,p(x)}{Z},\quad Z=\int_\Omega D(x)\,p(x)\,d\mu.&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;\gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for } |v| &lt; c.&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;L(v)=\frac{L_0}{\gamma(v)}.&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;, for instance:
* &lt;math&gt;D(x)=\gamma(x)&lt;/math&gt; (weights states by increasing &lt;math&gt;\gamma&lt;/math&gt;), or
* &lt;math&gt;D(x)=\frac{1}{\gamma(x)}&lt;/math&gt; (weights states by decreasing &lt;math&gt;\gamma&lt;/math&gt;),

provided &lt;math&gt;D(x)&lt;/math&gt; is strictly positive over the model’s domain and &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;.

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

=== Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy) ===
This brief example shows how &lt;math&gt;\pi&lt;/math&gt; naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical [[w:Buffon's needle problem|Buffon needle problem]] with needle length &lt;math&gt;L\le d&lt;/math&gt; and line spacing &lt;math&gt;d&lt;/math&gt;, the crossing probability is:
&lt;math&gt;P_{\mathrm{cross}}(L;d)=\frac{2L}{\pi d}.&lt;/math&gt;

If a rest-length &lt;math&gt;L_0&lt;/math&gt; undergoes Lorentz contraction &lt;math&gt;L(v)=L_0/\gamma(v)&lt;/math&gt; for &lt;math&gt;|v| &lt; c&lt;/math&gt;, then the same form gives:
&lt;math&gt;P_{\mathrm{cross}}(v)=\frac{2L_0}{\pi d\,\gamma(v)}.&lt;/math&gt;

Normalizing to the rest case yields a dimensionless sampling ratio:
&lt;math&gt;D_{\pi}(v):=\frac{P_{\mathrm{cross}}(v)}{P_{\mathrm{cross}}(0)}=\frac{1}{\gamma(v)}=\sqrt{1-\frac{v^2}{c^2}}.&lt;/math&gt;

This &lt;math&gt;D_{\pi}(v)&lt;/math&gt; is strictly positive on the domain &lt;math&gt;|v| &lt; c&lt;/math&gt; and can be used (illustratively) as a dilation field in the EPD transform:
&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D_{\pi}(v)\,dP}{\int_\Omega D_{\pi}(v)\,dP}.&lt;/math&gt;

== Theorem-style mathematics expansion ==

=== Definition (EPD reweighting operator) ===
Let &lt;math&gt;P&lt;/math&gt; be a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; and &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; measurable with &lt;math&gt;0&lt;\int_\Omega D\,dP&lt;\infty&lt;/math&gt;. Define &lt;math&gt;\mathrm{EPD}(P;D)=\widetilde{P}&lt;/math&gt; by:

&lt;math&gt;\widetilde{P}(A)=\frac{\int_A D\,dP}{\int_\Omega D\,dP}.&lt;/math&gt;

=== Theorem 1 (Normalization and positivity) ===
&lt;math&gt;\widetilde{P}&lt;/math&gt; is a probability measure. In particular:
# &lt;math&gt;\widetilde{P}(A)\ge 0&lt;/math&gt; for all measurable &lt;math&gt;A&lt;/math&gt;,
# &lt;math&gt;\widetilde{P}(\Omega)=1&lt;/math&gt;,
# If &lt;math&gt;P(A)=0&lt;/math&gt; then &lt;math&gt;\widetilde{P}(A)=0&lt;/math&gt; (absolute continuity preserved).

=== Theorem 2 (Expectation reweighting identity) ===
If &lt;math&gt;f&lt;/math&gt; is integrable under &lt;math&gt;\widetilde{P}&lt;/math&gt;, then

&lt;math&gt;\mathbb{E}_{\widetilde{P}}[f]=\frac{\mathbb{E}_P[D\,f]}{\mathbb{E}_P[D]}.&lt;/math&gt;

=== Theorem 3 (Composition / iteration rule) ===
Let &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D_1)&lt;/math&gt; and &lt;math&gt;\widehat{P}=\mathrm{EPD}(\widetilde{P};D_2)&lt;/math&gt;. Then

&lt;math&gt;\widehat{P}=\mathrm{EPD}(P;D_1D_2)&lt;/math&gt;

provided &lt;math&gt;\int_\Omega D_1D_2\,dP&lt;/math&gt; is finite and nonzero.

=== Theorem 4 (Fixed-point condition) ===
&lt;math&gt;P&lt;/math&gt; is a fixed point of EPD under dilation &lt;math&gt;D&lt;/math&gt; (i.e., &lt;math&gt;\mathrm{EPD}(P;D)=P&lt;/math&gt;) if and only if &lt;math&gt;D&lt;/math&gt; is constant &lt;math&gt;P&lt;/math&gt;-almost everywhere.

== Invariant quantities ==
EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:
* ratios of expectations of the form &lt;math&gt;\mathbb{E}_P[D f]/\mathbb{E}_P[D]&lt;/math&gt;,
* equivalence classes of observables that remain unchanged under specific families of dilations,
* fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

= Core axioms of Einstein Probability Dilation =

=== Axiom 1 — Probability primacy ===
Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

=== Axiom 2 — Positive dilation ===
Dilation fields are positive (&lt;math&gt;D&lt;/math&gt; is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

=== Axiom 3 — Iterative composability ===
Successive dilations compose multiplicatively (&lt;math&gt;D_1D_2&lt;/math&gt;), enabling multi-step (iterative) dynamics on measures.

= Falsifiability conditions =
A proposed EPD-based model should specify:
* the configuration space &lt;math&gt;\Omega&lt;/math&gt;,
* a baseline measure &lt;math&gt;P&lt;/math&gt;,
* an explicit dilation field &lt;math&gt;D&lt;/math&gt; (with parameters and domain),
* measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
* a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

'''Concrete falsifiability test (clustering example):''' Choose a baseline mock galaxy catalog (configuration space &lt;math&gt;\Omega&lt;/math&gt;), define an explicit strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt; using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed &lt;math&gt;\xi(r)&lt;/math&gt; over specified separation bins such as &lt;math&gt;r\le 60\,\mathrm{Mpc}/h&lt;/math&gt;). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

= Numerical simulation and iterative models =

== Simulation model description ==
In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure &lt;math&gt;P&lt;/math&gt; may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:
* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;, or
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing \xi(r) ===
This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;. The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

# generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
# define a strictly positive dilation field &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt; over galaxy positions,
# perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
# measure &lt;math&gt;\xi(r)&lt;/math&gt; using the properly normalized Landy–Szalay estimator,
# tune a small mini-grid over &lt;math&gt;(\sigma,\lambda)&lt;/math&gt; to minimize a simple mismatch score on small scales.

==== Key definitions ====
* Dilation field: &lt;math&gt;D(x)=\exp(\lambda\,\phi(x))&lt;/math&gt;, with &lt;math&gt;\lambda &gt; 0&lt;/math&gt; and &lt;math&gt;\phi(x)\ge 0&lt;/math&gt;.
* Example potential: &lt;math&gt;\phi(x)=\sum_k \exp\!\left(-\|x-s_k\|^2/(2\sigma^2)\right)&lt;/math&gt; using seed points &lt;math&gt;s_k&lt;/math&gt;.
* Landy–Szalay estimator (normalized counts):
  &lt;math&gt;\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}&lt;/math&gt;,
  where &lt;math&gt;DD,DR,RR&lt;/math&gt; are pair counts normalized by the total number of possible pairs in each catalog.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    &quot;&quot;&quot;
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i&lt;j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    &quot;&quot;&quot;
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i&lt;j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start &gt;= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    &quot;&quot;&quot;
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    &quot;&quot;&quot;
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    &quot;&quot;&quot;
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    &quot;&quot;&quot;
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    &quot;&quot;&quot;
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    &quot;&quot;&quot;
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = &quot;samushia_2013_CMASSDR11_xi_multipoles.dat&quot;
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 =&gt; centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple &quot;success&quot; on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    &quot;&quot;&quot;
    Mean squared error over bins with r &lt;= rmax (simple success criterion).
    &quot;&quot;&quot;
    m = r_centers &lt;= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score &lt; best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label=&quot;Mock baseline (uniform Poisson)&quot;)
plt.plot(r_centers, xi_best, marker='o',
         label=f&quot;Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})&quot;)
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label=&quot;Observed BOSS DR11 ξ₀(r) (reference)&quot;)
plt.axhline(0, linewidth=1)
plt.xlabel(&quot;r [Mpc/h]&quot;)
plt.ylabel(&quot;ξ(r) (monopole estimate)&quot;)
plt.title(&quot;EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)&quot;)
plt.legend()
plt.show()

print(&quot;=== Mini-grid search result ===&quot;)
print(f&quot;Best score (MSE on r&lt;=60): {best_score:.6e}&quot;)
print(f&quot;Best sigma: {best_sigma}  Best lambda: {best_lam}&quot;)
&lt;/syntaxhighlight&gt;

'''How to run:''' download the BOSS multipoles text file to your computer and set &lt;code&gt;path&lt;/code&gt; in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near &lt;math&gt;\xi\approx 0&lt;/math&gt;, while the EPD-resampled mock typically shows positive small-scale clustering.
== Scope and limitations ==
EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics,
* empirical confirmation without explicit predictions and tests.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==
{{Disclaimer|This section is speculative and exploratory in nature.}}

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19&lt;/ref&gt;

In particular, studies of &quot;Buffon deficits&quot; on Gaussian manifolds indicate that deviations from the classical flat-space Buffon probability may encode Gaussian curvature through small geometric corrections.

In flat geometry, the classical Buffon crossing probability approaches:

&lt;math&gt;
\frac{2}{\pi}
&lt;/math&gt;

for sufficiently small needles.

On curved manifolds, however, the crossing probability acquires curvature-dependent corrections. This motivates a speculative extension of Einstein Probability Dilation (EPD): spacetime curvature may not only influence probability distributions, but probability structure itself may participate in the emergence or modulation of geometric behavior.

Within this interpretation, curvature may act as a statistical weighting mechanism on quantum path behavior. Rather than treating quantum path integrals strictly as sums over equally admissible histories, spacetime geometry could modulate the statistical contribution of classes of paths through curvature-dependent weighting functions.&lt;ref&gt;Feynman, R. P., &amp; Hibbs, A. R. (1965). ''Quantum Mechanics and Path Integrals''. McGraw-Hill.&lt;/ref&gt;

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where:

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

represents the standard quantum probability amplitude,

&lt;math&gt;W(K)&lt;/math&gt;

represents a curvature-dependent geometric weighting factor,

and

&lt;math&gt;K&lt;/math&gt;

represents local Gaussian or spacetime curvature.

In this interpretation, the two probabilistic structures are not identical in mathematical character.

The Feynman factor

&lt;math&gt;e^{iS/\hbar}&lt;/math&gt;

is a complex quantum probability amplitude capable of interference and phase cancellation, whereas the curvature-weighting term behaves more like a real geometric measure acting on the statistical structure of path space itself.

Accordingly, EPD may not combine probabilities directly, but instead combine:

* a quantum interference structure,
* with a curvature-dependent probabilistic geometry.

The Buffon deficit framework suggests that curvature may appear as a probabilistic deviation from flat geometric behavior. Within EPD, this motivates the possibility that curvature-weighted probability measures could alter the statistical admissibility of quantum paths at extremely small scales.

If such curvature-weighting naturally suppresses pathological ultraviolet path contributions, the framework could potentially resemble a form of geometric renormalization in which divergence control emerges from geometric probability structure rather than purely algebraic subtraction procedures.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx 1+cKL^2+O(L^3)
&lt;/math&gt;

where:

&lt;math&gt;L&lt;/math&gt;

represents a microscopic path scale,

&lt;math&gt;K&lt;/math&gt;

represents local curvature,

and

&lt;math&gt;c&lt;/math&gt;

represents a proportionality constant.

In the flat-space limit:

&lt;math&gt;
K\rightarrow0
&lt;/math&gt;

ordinary quantum behavior would be recovered.

At present these ideas remain heuristic and conceptual. However, they suggest a possible bridge between:

* stochastic geometry,
* probabilistic curvature,
* quantum path integrals,
* and emergent spacetime structure.

=== Open Questions ===

* Can curvature-weighted path measures be formally derived?
* Can Buffon deficit relations be generalized to relativistic spacetime?
* Could curvature-weighting regulate ultraviolet divergences?
* What relationship exists between Gaussian curvature and spacetime curvature tensors?
* Could probabilistic geometry contribute to emergent spacetime models?

== Future directions ==

* develop canonical families of dilation fields and invariants,
* clarify “structure-from-measure” diagnostics,
* publish reproducible simulation notebooks and parameter sweeps,
* compare multiple dilation families under shared evaluation criteria.

== Status of the Framework ==
Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model. Several portions of the article discuss heuristic interpretations and proposed extensions which are not established physical theory.

The article is intended as an educational and exploratory investigation into possible relationships between probability structure, geometric curvature, and quantum path behavior.

== See also ==

* [[wikipedia:Buffon's needle problem|Buffon's needle problem]]
* [[wikipedia:Probability measure|Probability measure]]
* [[wikipedia:Cosmological principle|Cosmological principle]]
* [[wikipedia:Holographic principle|Holographic principle]]
* [[wikipedia:Fractal geometry|Fractal geometry]]
* [[wikipedia:Quantum field theory|Quantum field theory]]
* [[wikipedia:Langlands program|Langlands program]]

== References ==
&lt;references /&gt;

=== Further reading ===

==== Probability and measure theory ====

* [[wikipedia:Measure (mathematics)|Measure]]
* [[wikipedia:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[wikipedia:Probability theory|Probability theory]]

==== Computing and simulation ====

* [[wikipedia:Monte Carlo method|Monte Carlo method]]
* [[wikipedia:Importance sampling|Importance sampling]]

== Copyright and licensing ==
© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.

Earlier and related versions may also be available on OSF and arXiv under the licenses stated on those platforms. Where multiple versions exist, the license shown on the specific platform/version governs reuse of that version.</text>
      <sha1>qyf6pemvk276obk76yhu9qbbvvggu3l</sha1>
    </revision>
    <revision>
      <id>2810653</id>
      <parentid>2810651</parentid>
      <timestamp>2026-05-20T20:44:16Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Complete article edit.</comment>
      <origin>2810653</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8101" sha1="6t5h0wqc7xfv1gl3stw34jup70k24o5" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.

The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)=
\frac{\int_A D\,dP}
{\int_\Omega D\,dP}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\quad\text{for } |v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be a finite set, such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': keep all points and carry weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': draw a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing &lt;math&gt;\xi(r)&lt;/math&gt; ===

This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the monopole two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;.

{{Note|Unless observational datasets are explicitly supplied, this demonstration may be run using synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run with a synthetic target curve, the baseline mock catalog should remain close to &lt;math&gt;\xi\approx0&lt;/math&gt;, while the EPD-resampled mock may show enhanced small-scale clustering relative to the baseline mock catalog.

{{collapse top|Python demonstration code}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Python demonstration code moved here intentionally collapsed
# Full implementations are better maintained on GitHub or OSF
&lt;/syntaxhighlight&gt;

{{collapse bottom}}

== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x \, W(K)\, e^{iS/\hbar}
&lt;/math&gt;

where &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx1+cKL^2+O(L^3)
&lt;/math&gt;

At present these ideas remain heuristic and conceptual.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria.

== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>6t5h0wqc7xfv1gl3stw34jup70k24o5</sha1>
    </revision>
    <revision>
      <id>2810656</id>
      <parentid>2810653</parentid>
      <timestamp>2026-05-20T20:56:17Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>More editing.</comment>
      <origin>2810656</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10424" sha1="gp5fjxzcrwlj8b3sdu76hithrhiss8r" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.

The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)=
\frac{\int_A D\,dP}
{\int_\Omega D\,dP}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\quad\text{for } |v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing &lt;math&gt;\xi(r)&lt;/math&gt; ===

This demonstration connects the EPD reweighting framework to a standard observable in large-scale structure: the monopole two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;.

The objective is not a precision cosmology fit, but a reproducible and falsifiable demonstration of probabilistic reweighting effects under controlled conditions.

The demonstration pipeline is:

# generate a baseline mock galaxy catalog in a periodic box;
# define a strictly positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function;
# compare the transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; is a dilation-strength parameter;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative field over the configuration space.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are appropriately normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may be performed using synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, the baseline mock catalog should remain close to &lt;math&gt;\xi\approx0&lt;/math&gt;, while EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and simulation notebooks for EPD-style resampling demonstrations may be maintained on supplementary pages or external repositories.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}

== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>gp5fjxzcrwlj8b3sdu76hithrhiss8r</sha1>
    </revision>
    <revision>
      <id>2810657</id>
      <parentid>2810656</parentid>
      <timestamp>2026-05-20T21:04:23Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>/* Einstein Probability Dilation (EPD) */</comment>
      <origin>2810657</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11046" sha1="plz8s5x0ajjtnsgn2go65uw8uf1b5qq" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)=
\frac{\int_A D\,dP}
{\int_\Omega D\,dP}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\quad\text{for } |v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs and comparing &lt;math&gt;\xi(r)&lt;/math&gt; ===

This demonstration connects the EPD reweighting framework to a standard observable in large-scale structure: the monopole two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;.

The objective is not a precision cosmology fit, but a reproducible and falsifiable demonstration of probabilistic reweighting effects under controlled conditions.

The demonstration pipeline is:

# generate a baseline mock galaxy catalog in a periodic box;
# define a strictly positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function;
# compare the transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; is a dilation-strength parameter;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative field over the configuration space.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are appropriately normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may be performed using synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, the baseline mock catalog should remain close to &lt;math&gt;\xi\approx0&lt;/math&gt;, while EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and simulation notebooks for EPD-style resampling demonstrations may be maintained on supplementary pages or external repositories.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}

== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>plz8s5x0ajjtnsgn2go65uw8uf1b5qq</sha1>
    </revision>
    <revision>
      <id>2810658</id>
      <parentid>2810657</parentid>
      <timestamp>2026-05-20T21:08:46Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Simplify the simulation section</comment>
      <origin>2810658</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10653" sha1="3vf58q5vzrg50vc0r16haxmv5ol5109" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)=
\frac{\int_A D\,dP}
{\int_\Omega D\,dP}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\quad\text{for } |v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>3vf58q5vzrg50vc0r16haxmv5ol5109</sha1>
    </revision>
    <revision>
      <id>2810660</id>
      <parentid>2810658</parentid>
      <timestamp>2026-05-20T21:22:30Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>More edits.</comment>
      <origin>2810660</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10932" sha1="ktnfaebx5jirk3f0yb9laq7xourvei7" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)=
\frac{\int_A D\,dP}
{\int_\Omega D\,dP}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.


== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\quad\text{for } |v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>ktnfaebx5jirk3f0yb9laq7xourvei7</sha1>
    </revision>
    <revision>
      <id>2810662</id>
      <parentid>2810660</parentid>
      <timestamp>2026-05-20T21:27:33Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>New section Composition of Dilations</comment>
      <origin>2810662</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11995" sha1="s2phpbxirozt8ycgzhl6d4mpcqjm4uz" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation (EPD) =

== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;\mathbb{E}_P[f]=\int_\Omega f\,dP&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)=
\frac{\int_A D\,dP}
{\int_\Omega D\,dP}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.


== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
\quad\text{for } |v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2D_1\,dP}
{\int_\Omega D_2D_1\,dP}
&lt;/math&gt;

showing that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)\approx1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>s2phpbxirozt8ycgzhl6d4mpcqjm4uz</sha1>
    </revision>
    <revision>
      <id>2810694</id>
      <parentid>2810662</parentid>
      <timestamp>2026-05-21T00:18:51Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>formula edits for display.</comment>
      <origin>2810694</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12011" sha1="moe5tin01513wwe75lt04t5ta03q4ur" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation =
== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

where

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.


== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

showing that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>moe5tin01513wwe75lt04t5ta03q4ur</sha1>
    </revision>
    <revision>
      <id>2810695</id>
      <parentid>2810694</parentid>
      <timestamp>2026-05-21T00:28:20Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <origin>2810695</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12109" sha1="abducfszhrf00e98o05kf8gjyveqtqi" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

= Einstein Probability Dilation =
== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
&lt;math&gt;Z(P,D)=\int_\Omega D\,dP&lt;/math&gt;.
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==
Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with
&lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;,
define the '''EPD transform'''
&lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

where

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then
&lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;
where

&lt;math&gt;
\widetilde{p}(x)=
\frac{D(x)\,p(x)}{Z},
\quad
Z=\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:'''
the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.


== Example: Lorentz contraction as a positive dilation field ==
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

showing that sequential EPD transformations compose through multiplication of the dilation fields.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>abducfszhrf00e98o05kf8gjyveqtqi</sha1>
    </revision>
    <revision>
      <id>2810697</id>
      <parentid>2810695</parentid>
      <timestamp>2026-05-21T00:50:16Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>More edits.</comment>
      <origin>2810697</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13690" sha1="nb9bby2h81aaieoobr1srlk0c5n9yd0" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;

In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>nb9bby2h81aaieoobr1srlk0c5n9yd0</sha1>
    </revision>
    <revision>
      <id>2810699</id>
      <parentid>2810697</parentid>
      <timestamp>2026-05-21T01:19:24Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Add Entropy and iterative probability flow.</comment>
      <origin>2810699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15046" sha1="0xm2z99bjh7jchsrafl1qnbtymqyg40" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>0xm2z99bjh7jchsrafl1qnbtymqyg40</sha1>
    </revision>
    <revision>
      <id>2810701</id>
      <parentid>2810699</parentid>
      <timestamp>2026-05-21T01:22:38Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>A simple iterative interpretation</comment>
      <origin>2810701</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15912" sha1="hfzda16pehwtl0wpgu1id61ela0iqr6" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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      <text bytes="15711" sha1="4be3ei9fcijp6yct0so4qjwtdpnc008" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>k2vtog70wh0465j9x6uhkk7o12k1dlv</sha1>
    </revision>
    <revision>
      <id>2810716</id>
      <parentid>2810709</parentid>
      <timestamp>2026-05-21T02:28:35Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
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      <comment>Working toy experiment.</comment>
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      <text bytes="20862" sha1="b5am6mfgd5qcc5mlj9m4rvbf88rbjxs" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}

These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>b5am6mfgd5qcc5mlj9m4rvbf88rbjxs</sha1>
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    <revision>
      <id>2810717</id>
      <parentid>2810716</parentid>
      <timestamp>2026-05-21T02:32:01Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Install multi-peak localized field.</comment>
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      <text bytes="22750" sha1="b7u8tjxajn3xy4m306s8zrng0zxeka4" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}

These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of EPD transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative EPD dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}

These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of EPD transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative EPD dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of EPD transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* random attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect EPD to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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      <text bytes="26083" sha1="gz1lj44ji4o1skepb6s5uoaf5c3gsor" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}

These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of EPD transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative EPD dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of EPD transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* random attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect EPD to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.
== Qualitative classes of iterative EPD behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated EPD transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that repeated positive probability reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the EPD framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized EPD dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of EPD transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative EPD dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of EPD transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* random attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect EPD to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.
== Qualitative classes of iterative EPD behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated EPD transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that repeated positive probability reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the EPD framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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      <text bytes="26318" sha1="2zgyw6ks4ykiim0zk5vuj10df1shk24" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}

[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized EPD dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of EPD transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative EPD dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of EPD transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* random attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect EPD to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.
== Qualitative classes of iterative EPD behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated EPD transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that repeated positive probability reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the EPD framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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      <text bytes="26450" sha1="6g60uvifzup90ebr3maszk6iy1aa258" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Einstein Probability Dilation (EPD)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the EPD framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, EPD does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability.
The Einstein Buffon Process (EBP) refers to the probabilistic-geometric framework underlying Einstein Probability Dilation (EPD). EPD represents the proposed physical interpretation and extension of these probabilistic-geometric concepts into relativistic and quantum contexts.
A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== Probability dilation (reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''EPD transform''' &lt;math&gt;\widetilde{P}=\mathrm{EPD}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

EPD is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, EPD emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The EPD-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how EPD shifts probability mass toward regions with larger dilation weights while preserving normalization.
== Composition of dilations ==

An important structural property of EPD is that sequential dilations compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential EPD transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
showing that sequential EPD transformations compose through multiplication of the dilation fields.
This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.
== Fixed points and iterative dynamics ==

An important question in EPD concerns the long-term behavior of repeated probability dilations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive EPD transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the EPD transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;.More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.


More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect EPD to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the EPD framework.
== Entropy and iterative probability flow ==

Repeated EPD transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative probability dilation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect EPD to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative EPD transformations remains an open area for investigation.
== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated EPD transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the EPD update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative EPD behavior.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}

The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of EPD transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the EPD transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}

[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized EPD dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Oscillatory dilation fields ===

Another useful class of EPD transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive EPD iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative EPD dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of EPD transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative EPD dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of EPD transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated EPD iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect EPD to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.
== Qualitative classes of iterative EPD behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated EPD transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that repeated positive probability reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the EPD framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.
== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.
=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of EPD may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform EPD-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, EPD-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and limitations ==

EPD is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information.

Within the EPD framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

A schematic form may be written as:

&lt;math&gt;
\int \mathcal{D}x\,W(K)\,e^{iS/\hbar}
&lt;/math&gt;

where:

* &lt;math&gt;e^{iS/\hbar}&lt;/math&gt; represents the standard quantum probability amplitude;
* &lt;math&gt;W(K)&lt;/math&gt; represents a curvature-dependent weighting factor;
* &lt;math&gt;K&lt;/math&gt; represents local curvature.

A speculative curvature-weighting correction might take the approximate form:

&lt;math&gt;
W(K,L)
\approx
1+cKL^2+O(L^3)
&lt;/math&gt;

where:

* &lt;math&gt;L&lt;/math&gt; represents a microscopic path scale;
* &lt;math&gt;K&lt;/math&gt; represents local curvature;
* &lt;math&gt;c&lt;/math&gt; is a proportionality parameter.

At present these ideas remain heuristic and conceptual. They are included as exploratory possibilities rather than established physical theory.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.


== Status of the Framework ==

Einstein Probability Dilation (EPD) presently represents a speculative conceptual framework combining probabilistic geometry, relativistic interpretation, and stochastic path structures.

The framework has not been experimentally verified and presently exists as an exploratory mathematical and conceptual model.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Quantum field theory|Quantum field theory]]

== References ==

&lt;references/&gt;

== Copyright and licensing ==

© Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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    <title>Motivation and emotion/Book/2025/Coercion and therapeutic alliance</title>
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      <text bytes="29954" sha1="mmfkzb1q4gif1qcziwwgsz32fichphx" xml:space="preserve">{{title|Coercion and therapeutic alliance:&lt;br&gt;How do coercive practices in mental health care undermine trust and therapeutic relationships?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}

;Scenario

Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a 'difficult patient'. Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. 

Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings.

* What coercive practices has Sarah experienced?
* What factors put Sarah at a higher risk of experiencing coercive practices? 
* What barriers to good therapeutic relationships are experienced by Sarah?
* What could be done differently do improve her treatment outcomes?

{{RoundBoxBottom}}

{{expand}}

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What are therapeutic relationships and coercive practices?
* How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships?
* What are the impacts of coercive practices on therapeutic relationships and mental health outcomes?
* What factors can make someone vulnerable to coercive practices?
* What are the ethical implications of coercive practices, and how can they be overcome?

{{RoundBoxBottom}}

== Therapeutic relationships ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Defining therapeutic relationships ===
Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland &amp; Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018). Examples of therapeutic relationships in mental health settings include:
* Nurses tending to a psychiatric care patient (Ernstmeyer &amp; Christman, 2022).
* A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland &amp; Torrico, 2024).

=== Applying self-determination theory to therapeutic relationships ===
[[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci &amp; Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan &amp; Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012).
* Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider.
* Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle.
*Relatedness: You feel understood and cared for by your mental healthcare provider.
When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan &amp; Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance.  

=== Applying tripartite model of working alliance to therapeutic relationships ===
The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson &amp; Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward.

* Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties.
* Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. 
* Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. 

Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes.

[[File:Practitioner and patient.png|thumb|'''Figure 2:''' Coercive practices can be subtle and hard to spot.]]

== Coercive practices in mental health contexts ==
Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner &amp; Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). 

=== Defining coercive practices ===
Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames &amp; Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). 

'''Table 1.''' 

Examples of coercive practices  
{| class=&quot;wikitable&quot;
|+
!Restraint 
!Type of coercion
!Example
|-
|Physical
|Formal/explicit
|Use of physical force to restrain a patient
|-
|Mechanical
|Formal/explicit
|Use of physical tools  to restrain a patient
|-
|Chemical
|Formal/explicit
|Use of medication such as sedatives to control a patient's behaviour, rather than treating them
|-
|Environmental
|Formal/explicit
|Limiting access to environment through practices like seclusion 
|-
|Psychological
|Informal/implicit
|Using psychological pressures such as implicit threats or consequences to get patients to comply 
|}

&lt;quiz display=&quot;simple&quot;&gt;
{Coercive practices improve therapeutic relationships:
|type=&quot;()&quot;}
+ True
- False

{Coercive practices are always explicit:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

=== Impact of coercive practices on therapeutic relationships and mental health outcomes ===
Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes &amp; Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025).    

'''Table 2.''' 

Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust 
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
!Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance 
|-
|Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks.
|-
|Reduced engagement with mental health care. 
|Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management.  
|People may be unable to receive a diagnosis or to set mental health goals.
|-
|High incidence of trauma/trauma-related disorders, and loss of trust. 
|Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. 
|Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. 
|}

=== Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes ===
Coercive practices have negative impacts on anyone who experiences them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016).

Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships.     

Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland &amp; Torrico, 2024) (Sharma &amp; Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025).   

Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes &amp; Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025).

Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma &amp; Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). 

=== Ethical implications of coercive practices ===
Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner &amp; Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames &amp; Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified  in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner &amp; Braun, 2023). 

== Overcoming coercive practices ==
Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018).  

==Conclusion==
In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. 

The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. 

Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. 

==See also==
* [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018)
* [[wikipedia:Coercion|Coercion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia)
* [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia)
* [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia)
* [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia)
* [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia)
* [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia)
* [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia)

==References==
{{Hanging indent|1=

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}}

==External links==
* [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com)
* [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing)

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==Overview==
{{RoundBoxTop|theme=3}}

;Scenario

Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a 'difficult patient'. Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. 

Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings.

* What coercive practices has Sarah experienced?
* What factors put Sarah at a higher risk of experiencing coercive practices? 
* What barriers to good therapeutic relationships are experienced by Sarah?
* What could be done differently do improve her treatment outcomes?

{{RoundBoxBottom}}

{{expand}}

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What are therapeutic relationships and coercive practices?
* How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships?
* What are the impacts of coercive practices on therapeutic relationships and mental health outcomes?
* What factors can make someone vulnerable to coercive practices?
* What are the ethical implications of coercive practices, and how can they be overcome?

{{RoundBoxBottom}}

== Therapeutic relationships ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Defining therapeutic relationships ===
Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland &amp; Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018). Examples of therapeutic relationships in mental health settings include:
* Nurses tending to a psychiatric care patient (Ernstmeyer &amp; Christman, 2022).
* A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland &amp; Torrico, 2024).

=== Applying self-determination theory to therapeutic relationships ===
[[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci &amp; Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan &amp; Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012).
* Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider.
* Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle.
*Relatedness: You feel understood and cared for by your mental healthcare provider.
When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan &amp; Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance.  

=== Applying tripartite model of working alliance to therapeutic relationships ===
The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson &amp; Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward.

* Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties.
* Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. 
* Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. 

Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes.

== Coercive practices in mental health contexts ==
Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner &amp; Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). 

=== Defining coercive practices ===
Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames &amp; Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). 

'''Table 1.''' 

Examples of coercive practices  
{| class=&quot;wikitable&quot;
|+
!Restraint 
!Type of coercion
!Example
|-
|Physical
|Formal/explicit
|Use of physical force to restrain a patient
|-
|Mechanical
|Formal/explicit
|Use of physical tools  to restrain a patient
|-
|Chemical
|Formal/explicit
|Use of medication such as sedatives to control a patient's behaviour, rather than treating them
|-
|Environmental
|Formal/explicit
|Limiting access to environment through practices like seclusion 
|-
|Psychological
|Informal/implicit
|Using psychological pressures such as implicit threats or consequences to get patients to comply 
|}

&lt;quiz display=&quot;simple&quot;&gt;
{Coercive practices improve therapeutic relationships:
|type=&quot;()&quot;}
+ True
- False

{Coercive practices are always explicit:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

=== Impact of coercive practices on therapeutic relationships and mental health outcomes ===
Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes &amp; Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025).    

'''Table 2.''' 

Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust 
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
!Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance 
|-
|Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks.
|-
|Reduced engagement with mental health care. 
|Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management.  
|People may be unable to receive a diagnosis or to set mental health goals.
|-
|High incidence of trauma/trauma-related disorders, and loss of trust. 
|Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. 
|Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. 
|}

=== Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes ===
Coercive practices have negative impacts on anyone who experiences them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016).

Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships.     

Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland &amp; Torrico, 2024) (Sharma &amp; Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025).   

Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes &amp; Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025).

Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma &amp; Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). 

=== Ethical implications of coercive practices ===
Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner &amp; Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames &amp; Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified  in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner &amp; Braun, 2023). 

== Overcoming coercive practices ==
Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018).  

==Conclusion==
In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. 

The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. 

Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. 

==See also==
* [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018)
* [[wikipedia:Coercion|Coercion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia)
* [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia)
* [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia)
* [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia)
* [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia)
* [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia)
* [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia)

==References==
{{Hanging indent|1=

Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., &amp; Alegría, M. (2015). Patient-Provider therapeutic alliance contributes to patient activation in community mental health clinics. Administration and Policy in Mental Health and Mental Health Services Research, 44(4), 431–440. https://doi.org/10.1007/s10488-015-0655-8

Beames, L., &amp; Onwumere, J. (2021). Risk factors associated with use of coercive practices in adult inpatient mental health patients: A systematic review. Journal of Psychiatric and Mental Health Nursing, 29(2). https://doi.org/10.1111/jpm.12757

Carmichael, C., Schiffler, T., Smith, L., Moudatsou, M., Tabaki, I., Doñate-Martínez, A., Alhambra-Borrás, T., Kouvari, M., Karnaki, P., Gil-Salmeron, A., &amp; Grabovac, I. (2023). Barriers and facilitators to health care access for people experiencing homelessness in four european countries: An exploratory qualitative study. International Journal for Equity in Health, 22(1), 206. https://doi.org/10.1186/s12939-023-02011-4

Chieze, M., Clavien, C., Kaiser, S., &amp; Hurst, S. (2021). Coercive measures in psychiatry: A review of ethical arguments. Frontiers in Psychiatry, 12(12). Pubmed Central. https://doi.org/10.3389/fpsyt.2021.790886

Critical Intelligence unit. (2025). Evidence check reducing restrictive practices. https://aci.health.nsw.gov.au/__data/assets/pdf_file/0004/1004377/Evidence-check-Reducing-restrictive-and-coercive-practices.pdf

Deci, E. L., &amp; Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27(1), 23–40. https://selfdeterminationtheory.org/SDT/documents/2004_DeciVansteenkiste_SDTandBasicNeedSatisfaction.pdf

Ernstmeyer, K., &amp; Christman, E. (2022). Therapeutic communication and the nurse-client relationship. In www.ncbi.nlm.nih.gov. Chippewa Valley Technical College. https://www.ncbi.nlm.nih.gov/books/NBK590036/ (Original work published 2025)

Faissner, M., &amp; Braun, E. (2023). The ethics of coercion in mental healthcare: The role of structural racism. Journal of Medical Ethics, 50(7). https://doi.org/10.1136/jme-2023-108984

Gilburt, H., Rose, D., &amp; Slade, M. (2008). The importance of relationships in mental health care: A qualitative study of service users’ experiences of psychiatric hospital admission in the UK. BMC Health Services Research, 8(1). https://doi.org/10.1186/1472-6963-8-92

Gooding, P., Mcsherry, B., Roper, C., &amp; Grey, F. (2018). Alternatives to coercion in mental health settings: A literature review. https://socialequity.unimelb.edu.au/__data/assets/pdf_file/0012/2898525/Alternatives-to-Coercion-Literature-Review-Melbourne-Social-Equity-Institute.pdf

Hamovitch, E. K., Choy-Brown, M., &amp; Stanhope, V. (2018). Person-Centered Care and the Therapeutic Alliance. Community Mental Health Journal, 54(7), 951–958. Springer Nature Link. https://doi.org/10.1007/s10597-018-0295-z

Hem, M. H., Gjerberg, E., Husum, T. L., &amp; Pedersen, R. (2018). Ethical challenges when using coercion in mental healthcare: A systematic literature review. Nursing Ethics, 25(1), 92–110. Sage Journals. https://doi.org/10.1177/0969733016629770

Iversen, H. W., Riley, H., Råbu, M., &amp; Lorem, G. F. (2025). Building and sustaining therapeutic relationships across treatment settings: A qualitative study of how patients navigate the group dynamics of mental healthcare. BMC Psychiatry, 25(1). https://doi.org/10.1186/s12888-025-06874-5

Johnson, L. N., &amp; Wright, D. W. (2002). Revisiting bordin’s theory on the therapeutic alliance: Implications for family therapy. Contemporary Family Therapy, 24(2), 257–269. https://doi.org/10.1023/a:1015395223978

Kornhaber, R., Walsh, K., Duff, J., &amp; Walker, K. (2016). Enhancing adult therapeutic interpersonal relationships in the acute health care setting: An integrative review. Journal of Multidisciplinary Healthcare, 9(14), 537–546. Pubmed Central. https://doi.org/10.2147/JMDH.S116957

Newton-Howes, G., &amp; Mullen, R. (2011). Coercion in psychiatric care: Systematic review of correlates and themes. Psychiatric Services, 62(5), 465–470. https://doi.org/10.1176/ps.62.5.pss6205_0465

Ng, J. Y. Y., Ntoumanis, N., Thøgersen-Ntoumani, C., Deci, E. L., Ryan, R. M., Duda, J. L., &amp; Williams, G. C. (2012). Self-Determination theory applied to health contexts. Perspectives on Psychological Science, 7(4), 325–340. https://doi.org/10.1177/1745691612447309

Norcross, J. C. (2010). The therapeutic relationship. Psycnet.apa.org; American Psychological Association. https://doi.org/10.1037/12075-004

Opland, C., &amp; Torrico, T. J. (2024, October 6). Psychotherapy and therapeutic relationship. National Library of Medicine; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK608012/

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryan, R. M., &amp; Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753

Sashidharan, S. P., Mezzina, R., &amp; Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350

Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version

Sharma, N., &amp; Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/

Stanhope, V., Marcus, S., &amp; Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183

Wostry, F., Hahn, S., &amp; Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012

}}

==External links==
* [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com)
* [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Control]]
[[Category:Motivation and emotion/Book/Psychotherapy]]
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==Overview==
{{RoundBoxTop|theme=3}}
;Scenario
[[File:Healthcare professional writing.jpg|thumb|225px|'''Figure 1'''. Patients may feel pressured into complying by healthcare providers.]]
Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a 'difficult patient'. Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. 

Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings.

* What coercive practices has Sarah experienced?
* What factors put Sarah at a higher risk of experiencing coercive practices? 
* What barriers to good therapeutic relationships are experienced by Sarah?
* What could be done differently do improve her treatment outcomes?

{{RoundBoxBottom}}

{{expand}}

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What are therapeutic relationships and coercive practices?
* How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships?
* What are the impacts of coercive practices on therapeutic relationships and mental health outcomes?
* What factors can make someone vulnerable to coercive practices?
* What are the ethical implications of coercive practices, and how can they be overcome?

{{RoundBoxBottom}}

== Therapeutic relationships ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Defining therapeutic relationships ===
Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland &amp; Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018). Examples of therapeutic relationships in mental health settings include:
* Nurses tending to a psychiatric care patient (Ernstmeyer &amp; Christman, 2022).
* A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland &amp; Torrico, 2024).

=== Applying self-determination theory to therapeutic relationships ===
[[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci &amp; Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan &amp; Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012).
* Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider.
* Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle.
*Relatedness: You feel understood and cared for by your mental healthcare provider.
When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan &amp; Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance.  

=== Applying tripartite model of working alliance to therapeutic relationships ===
The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson &amp; Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward.

* Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties.
* Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. 
* Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. 

Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes.

== Coercive practices in mental health contexts ==
Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner &amp; Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). 

=== Defining coercive practices ===
Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames &amp; Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). 

'''Table 1.''' 

Examples of coercive practices  
{| class=&quot;wikitable&quot;
|+
!Restraint 
!Type of coercion
!Example
|-
|Physical
|Formal/explicit
|Use of physical force to restrain a patient
|-
|Mechanical
|Formal/explicit
|Use of physical tools  to restrain a patient
|-
|Chemical
|Formal/explicit
|Use of medication such as sedatives to control a patient's behaviour, rather than treating them
|-
|Environmental
|Formal/explicit
|Limiting access to environment through practices like seclusion 
|-
|Psychological
|Informal/implicit
|Using psychological pressures such as implicit threats or consequences to get patients to comply 
|}

&lt;quiz display=&quot;simple&quot;&gt;
{Coercive practices improve therapeutic relationships:
|type=&quot;()&quot;}
+ True
- False

{Coercive practices are always explicit:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

=== Impact of coercive practices on therapeutic relationships and mental health outcomes ===
Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes &amp; Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025).    

'''Table 2.''' 

Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust 
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
!Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance 
|-
|Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks.
|-
|Reduced engagement with mental health care. 
|Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management.  
|People may be unable to receive a diagnosis or to set mental health goals.
|-
|High incidence of trauma/trauma-related disorders, and loss of trust. 
|Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. 
|Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. 
|}

=== Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes ===
Coercive practices have negative impacts on anyone who experiences them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016).

Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships.     

Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland &amp; Torrico, 2024) (Sharma &amp; Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025).   

Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes &amp; Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025).

Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma &amp; Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). 

=== Ethical implications of coercive practices ===
Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner &amp; Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames &amp; Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified  in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner &amp; Braun, 2023). 

== Overcoming coercive practices ==
Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018).  

==Conclusion==
In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. 

The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. 

Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. 

==See also==
* [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018)
* [[wikipedia:Coercion|Coercion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia)
* [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia)
* [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia)
* [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia)
* [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia)
* [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia)
* [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia)

==References==
{{Hanging indent|1=

Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., &amp; Alegría, M. (2015). Patient-Provider therapeutic alliance contributes to patient activation in community mental health clinics. Administration and Policy in Mental Health and Mental Health Services Research, 44(4), 431–440. https://doi.org/10.1007/s10488-015-0655-8

Beames, L., &amp; Onwumere, J. (2021). Risk factors associated with use of coercive practices in adult inpatient mental health patients: A systematic review. Journal of Psychiatric and Mental Health Nursing, 29(2). https://doi.org/10.1111/jpm.12757

Carmichael, C., Schiffler, T., Smith, L., Moudatsou, M., Tabaki, I., Doñate-Martínez, A., Alhambra-Borrás, T., Kouvari, M., Karnaki, P., Gil-Salmeron, A., &amp; Grabovac, I. (2023). Barriers and facilitators to health care access for people experiencing homelessness in four european countries: An exploratory qualitative study. International Journal for Equity in Health, 22(1), 206. https://doi.org/10.1186/s12939-023-02011-4

Chieze, M., Clavien, C., Kaiser, S., &amp; Hurst, S. (2021). Coercive measures in psychiatry: A review of ethical arguments. Frontiers in Psychiatry, 12(12). Pubmed Central. https://doi.org/10.3389/fpsyt.2021.790886

Critical Intelligence unit. (2025). Evidence check reducing restrictive practices. https://aci.health.nsw.gov.au/__data/assets/pdf_file/0004/1004377/Evidence-check-Reducing-restrictive-and-coercive-practices.pdf

Deci, E. L., &amp; Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27(1), 23–40. https://selfdeterminationtheory.org/SDT/documents/2004_DeciVansteenkiste_SDTandBasicNeedSatisfaction.pdf

Ernstmeyer, K., &amp; Christman, E. (2022). Therapeutic communication and the nurse-client relationship. In www.ncbi.nlm.nih.gov. Chippewa Valley Technical College. https://www.ncbi.nlm.nih.gov/books/NBK590036/ (Original work published 2025)

Faissner, M., &amp; Braun, E. (2023). The ethics of coercion in mental healthcare: The role of structural racism. Journal of Medical Ethics, 50(7). https://doi.org/10.1136/jme-2023-108984

Gilburt, H., Rose, D., &amp; Slade, M. (2008). The importance of relationships in mental health care: A qualitative study of service users’ experiences of psychiatric hospital admission in the UK. BMC Health Services Research, 8(1). https://doi.org/10.1186/1472-6963-8-92

Gooding, P., Mcsherry, B., Roper, C., &amp; Grey, F. (2018). Alternatives to coercion in mental health settings: A literature review. https://socialequity.unimelb.edu.au/__data/assets/pdf_file/0012/2898525/Alternatives-to-Coercion-Literature-Review-Melbourne-Social-Equity-Institute.pdf

Hamovitch, E. K., Choy-Brown, M., &amp; Stanhope, V. (2018). Person-Centered Care and the Therapeutic Alliance. Community Mental Health Journal, 54(7), 951–958. Springer Nature Link. https://doi.org/10.1007/s10597-018-0295-z

Hem, M. H., Gjerberg, E., Husum, T. L., &amp; Pedersen, R. (2018). Ethical challenges when using coercion in mental healthcare: A systematic literature review. Nursing Ethics, 25(1), 92–110. Sage Journals. https://doi.org/10.1177/0969733016629770

Iversen, H. W., Riley, H., Råbu, M., &amp; Lorem, G. F. (2025). Building and sustaining therapeutic relationships across treatment settings: A qualitative study of how patients navigate the group dynamics of mental healthcare. BMC Psychiatry, 25(1). https://doi.org/10.1186/s12888-025-06874-5

Johnson, L. N., &amp; Wright, D. W. (2002). Revisiting bordin’s theory on the therapeutic alliance: Implications for family therapy. Contemporary Family Therapy, 24(2), 257–269. https://doi.org/10.1023/a:1015395223978

Kornhaber, R., Walsh, K., Duff, J., &amp; Walker, K. (2016). Enhancing adult therapeutic interpersonal relationships in the acute health care setting: An integrative review. Journal of Multidisciplinary Healthcare, 9(14), 537–546. Pubmed Central. https://doi.org/10.2147/JMDH.S116957

Newton-Howes, G., &amp; Mullen, R. (2011). Coercion in psychiatric care: Systematic review of correlates and themes. Psychiatric Services, 62(5), 465–470. https://doi.org/10.1176/ps.62.5.pss6205_0465

Ng, J. Y. Y., Ntoumanis, N., Thøgersen-Ntoumani, C., Deci, E. L., Ryan, R. M., Duda, J. L., &amp; Williams, G. C. (2012). Self-Determination theory applied to health contexts. Perspectives on Psychological Science, 7(4), 325–340. https://doi.org/10.1177/1745691612447309

Norcross, J. C. (2010). The therapeutic relationship. Psycnet.apa.org; American Psychological Association. https://doi.org/10.1037/12075-004

Opland, C., &amp; Torrico, T. J. (2024, October 6). Psychotherapy and therapeutic relationship. National Library of Medicine; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK608012/

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryan, R. M., &amp; Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753

Sashidharan, S. P., Mezzina, R., &amp; Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350

Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version

Sharma, N., &amp; Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/

Stanhope, V., Marcus, S., &amp; Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183

Wostry, F., Hahn, S., &amp; Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012

}}

==External links==
* [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com)
* [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing)

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__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
;Scenario
[[File:Healthcare professional writing.jpg|thumb|225px|'''Figure 1'''. Patients may feel pressured into complying by healthcare providers.]]
Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a 'difficult patient'. Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. 

Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings.

* What coercive practices has Sarah experienced?
* What factors put Sarah at a higher risk of experiencing coercive practices? 
* What barriers to good therapeutic relationships are experienced by Sarah?
* What could be done differently do improve her treatment outcomes?

{{RoundBoxBottom}}

{{expand}}

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What are therapeutic relationships and coercive practices?
* How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships?
* What are the impacts of coercive practices on therapeutic relationships and mental health outcomes?
* What factors can make someone vulnerable to coercive practices?
* What are the ethical implications of coercive practices, and how can they be overcome?

{{RoundBoxBottom}}

== Therapeutic relationships ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Defining therapeutic relationships ===
Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland &amp; Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018). Examples of therapeutic relationships in mental health settings include:
* Nurses tending to a psychiatric care patient (Ernstmeyer &amp; Christman, 2022).
* A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland &amp; Torrico, 2024).

=== Applying self-determination theory to therapeutic relationships ===
[[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci &amp; Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan &amp; Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012).
* Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider.
* Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle.
*Relatedness: You feel understood and cared for by your mental healthcare provider.
When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan &amp; Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance.  

=== Applying tripartite model of working alliance to therapeutic relationships ===
The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson &amp; Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward.

* Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties.
* Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. 
* Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. 

Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes.

[[File:Health professional in scrubs enjoying coffee while using a smartphone in a casual work setting during morning hours.jpg|thumb|'''Figure 2.''' Coercive practices can be subtle and hard to spot.]

== Coercive practices in mental health contexts ==
Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner &amp; Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). 

=== Defining coercive practices ===
Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames &amp; Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). 

'''Table 1.''' 

Examples of coercive practices  
{| class=&quot;wikitable&quot;
|+
!Restraint 
!Type of coercion
!Example
|-
|Physical
|Formal/explicit
|Use of physical force to restrain a patient
|-
|Mechanical
|Formal/explicit
|Use of physical tools  to restrain a patient
|-
|Chemical
|Formal/explicit
|Use of medication such as sedatives to control a patient's behaviour, rather than treating them
|-
|Environmental
|Formal/explicit
|Limiting access to environment through practices like seclusion 
|-
|Psychological
|Informal/implicit
|Using psychological pressures such as implicit threats or consequences to get patients to comply 
|}

&lt;quiz display=&quot;simple&quot;&gt;
{Coercive practices improve therapeutic relationships:
|type=&quot;()&quot;}
+ True
- False

{Coercive practices are always explicit:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

=== Impact of coercive practices on therapeutic relationships and mental health outcomes ===
Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes &amp; Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025).    

'''Table 2.''' 

Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust 
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
!Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance 
|-
|Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks.
|-
|Reduced engagement with mental health care. 
|Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management.  
|People may be unable to receive a diagnosis or to set mental health goals.
|-
|High incidence of trauma/trauma-related disorders, and loss of trust. 
|Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. 
|Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. 
|}

=== Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes ===
Coercive practices have negative impacts on anyone who experiences them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016).

Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships.     

Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland &amp; Torrico, 2024) (Sharma &amp; Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025).   

Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes &amp; Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025).

Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma &amp; Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). 

=== Ethical implications of coercive practices ===
Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner &amp; Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames &amp; Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified  in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner &amp; Braun, 2023). 

== Overcoming coercive practices ==
Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018).  

==Conclusion==
In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. 

The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. 

Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. 

==See also==
* [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018)
* [[wikipedia:Coercion|Coercion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia)
* [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia)
* [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia)
* [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia)
* [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia)
* [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia)
* [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia)

==References==
{{Hanging indent|1=

Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., &amp; Alegría, M. (2015). Patient-Provider therapeutic alliance contributes to patient activation in community mental health clinics. Administration and Policy in Mental Health and Mental Health Services Research, 44(4), 431–440. https://doi.org/10.1007/s10488-015-0655-8

Beames, L., &amp; Onwumere, J. (2021). Risk factors associated with use of coercive practices in adult inpatient mental health patients: A systematic review. Journal of Psychiatric and Mental Health Nursing, 29(2). https://doi.org/10.1111/jpm.12757

Carmichael, C., Schiffler, T., Smith, L., Moudatsou, M., Tabaki, I., Doñate-Martínez, A., Alhambra-Borrás, T., Kouvari, M., Karnaki, P., Gil-Salmeron, A., &amp; Grabovac, I. (2023). Barriers and facilitators to health care access for people experiencing homelessness in four european countries: An exploratory qualitative study. International Journal for Equity in Health, 22(1), 206. https://doi.org/10.1186/s12939-023-02011-4

Chieze, M., Clavien, C., Kaiser, S., &amp; Hurst, S. (2021). Coercive measures in psychiatry: A review of ethical arguments. Frontiers in Psychiatry, 12(12). Pubmed Central. https://doi.org/10.3389/fpsyt.2021.790886

Critical Intelligence unit. (2025). Evidence check reducing restrictive practices. https://aci.health.nsw.gov.au/__data/assets/pdf_file/0004/1004377/Evidence-check-Reducing-restrictive-and-coercive-practices.pdf

Deci, E. L., &amp; Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27(1), 23–40. https://selfdeterminationtheory.org/SDT/documents/2004_DeciVansteenkiste_SDTandBasicNeedSatisfaction.pdf

Ernstmeyer, K., &amp; Christman, E. (2022). Therapeutic communication and the nurse-client relationship. In www.ncbi.nlm.nih.gov. Chippewa Valley Technical College. https://www.ncbi.nlm.nih.gov/books/NBK590036/ (Original work published 2025)

Faissner, M., &amp; Braun, E. (2023). The ethics of coercion in mental healthcare: The role of structural racism. Journal of Medical Ethics, 50(7). https://doi.org/10.1136/jme-2023-108984

Gilburt, H., Rose, D., &amp; Slade, M. (2008). The importance of relationships in mental health care: A qualitative study of service users’ experiences of psychiatric hospital admission in the UK. BMC Health Services Research, 8(1). https://doi.org/10.1186/1472-6963-8-92

Gooding, P., Mcsherry, B., Roper, C., &amp; Grey, F. (2018). Alternatives to coercion in mental health settings: A literature review. https://socialequity.unimelb.edu.au/__data/assets/pdf_file/0012/2898525/Alternatives-to-Coercion-Literature-Review-Melbourne-Social-Equity-Institute.pdf

Hamovitch, E. K., Choy-Brown, M., &amp; Stanhope, V. (2018). Person-Centered Care and the Therapeutic Alliance. Community Mental Health Journal, 54(7), 951–958. Springer Nature Link. https://doi.org/10.1007/s10597-018-0295-z

Hem, M. H., Gjerberg, E., Husum, T. L., &amp; Pedersen, R. (2018). Ethical challenges when using coercion in mental healthcare: A systematic literature review. Nursing Ethics, 25(1), 92–110. Sage Journals. https://doi.org/10.1177/0969733016629770

Iversen, H. W., Riley, H., Råbu, M., &amp; Lorem, G. F. (2025). Building and sustaining therapeutic relationships across treatment settings: A qualitative study of how patients navigate the group dynamics of mental healthcare. BMC Psychiatry, 25(1). https://doi.org/10.1186/s12888-025-06874-5

Johnson, L. N., &amp; Wright, D. W. (2002). Revisiting bordin’s theory on the therapeutic alliance: Implications for family therapy. Contemporary Family Therapy, 24(2), 257–269. https://doi.org/10.1023/a:1015395223978

Kornhaber, R., Walsh, K., Duff, J., &amp; Walker, K. (2016). Enhancing adult therapeutic interpersonal relationships in the acute health care setting: An integrative review. Journal of Multidisciplinary Healthcare, 9(14), 537–546. Pubmed Central. https://doi.org/10.2147/JMDH.S116957

Newton-Howes, G., &amp; Mullen, R. (2011). Coercion in psychiatric care: Systematic review of correlates and themes. Psychiatric Services, 62(5), 465–470. https://doi.org/10.1176/ps.62.5.pss6205_0465

Ng, J. Y. Y., Ntoumanis, N., Thøgersen-Ntoumani, C., Deci, E. L., Ryan, R. M., Duda, J. L., &amp; Williams, G. C. (2012). Self-Determination theory applied to health contexts. Perspectives on Psychological Science, 7(4), 325–340. https://doi.org/10.1177/1745691612447309

Norcross, J. C. (2010). The therapeutic relationship. Psycnet.apa.org; American Psychological Association. https://doi.org/10.1037/12075-004

Opland, C., &amp; Torrico, T. J. (2024, October 6). Psychotherapy and therapeutic relationship. National Library of Medicine; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK608012/

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryan, R. M., &amp; Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753

Sashidharan, S. P., Mezzina, R., &amp; Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350

Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version

Sharma, N., &amp; Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/

Stanhope, V., Marcus, S., &amp; Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183

Wostry, F., Hahn, S., &amp; Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012

}}

==External links==
* [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com)
* [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing)

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__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
;Scenario
[[File:Healthcare professional writing.jpg|thumb|225px|'''Figure 1'''.Patients may feel pressured into complying by healthcare providers.]]
Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a &quot;difficult patient&quot;. Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. 

Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings (see Figure 1).

* What coercive practices has Sarah experienced?
* What factors put Sarah at a higher risk of experiencing coercive practices? 
* What barriers to good therapeutic relationships are experienced by Sarah?
* What could be done differently do improve her treatment outcomes?

{{RoundBoxBottom}}

{{expand}}

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* What are therapeutic relationships and coercive practices?
* How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships?
* What are the impacts of coercive practices on therapeutic relationships and mental health outcomes?
* What factors can make someone vulnerable to coercive practices?
* What are the ethical implications of coercive practices, and how can they be overcome?

{{RoundBoxBottom}}

== Therapeutic relationships ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Defining therapeutic relationships ===
[[File:Health professional in scrubs enjoying coffee while using a smartphone in a casual work setting during morning hours.jpg|thumb|225px|'''Figure 2.''' Coercive practices can be subtle and hard to spot.]]

Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland &amp; Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018) (see Figure 2). Examples of therapeutic relationships in mental health settings include:
* Nurses tending to a psychiatric care patient (Ernstmeyer &amp; Christman, 2022).
* A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland &amp; Torrico, 2024).

=== Applying self-determination theory to therapeutic relationships ===
[[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci &amp; Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan &amp; Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012).
* Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider.
* Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle.
*Relatedness: You feel understood and cared for by your mental healthcare provider.
When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan &amp; Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance.  

=== Applying tripartite model of working alliance to therapeutic relationships ===
The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson &amp; Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward.

* Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties.
* Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. 
* Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. 

Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes.

== Coercive practices in mental health contexts ==
Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner &amp; Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018).

=== Defining coercive practices ===
Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames &amp; Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021).

'''Table 1.''' 

Examples of coercive practices  
{| class=&quot;wikitable&quot;
|+
!Restraint 
!Type of coercion
!Example
|-
|Physical
|Formal/explicit
|Use of physical force to restrain a patient
|-
|Mechanical
|Formal/explicit
|Use of physical tools  to restrain a patient
|-
|Chemical
|Formal/explicit
|Use of medication such as sedatives to control a patient's behaviour, rather than treating them
|-
|Environmental
|Formal/explicit
|Limiting access to environment through practices like seclusion 
|-
|Psychological
|Informal/implicit
|Using psychological pressures such as implicit threats or consequences to get patients to comply 
|}

&lt;quiz display=&quot;simple&quot;&gt;
{Coercive practices improve therapeutic relationships:
|type=&quot;()&quot;}
+ True
- False

{Coercive practices are always explicit:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

=== Impact of coercive practices on therapeutic relationships and mental health outcomes ===
Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes &amp; Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025).    

'''Table 2.''' 

Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust 
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
!Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance 
|-
|Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks.
|-
|Reduced engagement with mental health care. 
|Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management.  
|People may be unable to receive a diagnosis or to set mental health goals.
|-
|High incidence of trauma/trauma-related disorders, and loss of trust. 
|Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. 
|Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. 
|}

=== Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes ===
Coercive practices have negative impacts on those who experience them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016).

Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships.     

Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland &amp; Torrico, 2024) (Sharma &amp; Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025).   

Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes &amp; Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025).

Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma &amp; Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). 

=== Ethical implications of coercive practices ===
Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner &amp; Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames &amp; Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified  in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner &amp; Braun, 2023). 

== Overcoming coercive practices ==
Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018).  

==Conclusion==
In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. 

The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. 

Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. 

==See also==
* [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018)
* [[wikipedia:Coercion|Coercion]] (Wikipedia)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia)
* [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia)
* [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia)
* [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia)
* [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia)
* [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia)
* [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia)

==References==
{{Hanging indent|1=

Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., &amp; Alegría, M. (2015). Patient-Provider therapeutic alliance contributes to patient activation in community mental health clinics. Administration and Policy in Mental Health and Mental Health Services Research, 44(4), 431–440. https://doi.org/10.1007/s10488-015-0655-8

Beames, L., &amp; Onwumere, J. (2021). Risk factors associated with use of coercive practices in adult inpatient mental health patients: A systematic review. Journal of Psychiatric and Mental Health Nursing, 29(2). https://doi.org/10.1111/jpm.12757

Carmichael, C., Schiffler, T., Smith, L., Moudatsou, M., Tabaki, I., Doñate-Martínez, A., Alhambra-Borrás, T., Kouvari, M., Karnaki, P., Gil-Salmeron, A., &amp; Grabovac, I. (2023). Barriers and facilitators to health care access for people experiencing homelessness in four european countries: An exploratory qualitative study. International Journal for Equity in Health, 22(1), 206. https://doi.org/10.1186/s12939-023-02011-4

Chieze, M., Clavien, C., Kaiser, S., &amp; Hurst, S. (2021). Coercive measures in psychiatry: A review of ethical arguments. Frontiers in Psychiatry, 12(12). Pubmed Central. https://doi.org/10.3389/fpsyt.2021.790886

Critical Intelligence unit. (2025). Evidence check reducing restrictive practices. https://aci.health.nsw.gov.au/__data/assets/pdf_file/0004/1004377/Evidence-check-Reducing-restrictive-and-coercive-practices.pdf

Deci, E. L., &amp; Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27(1), 23–40. https://selfdeterminationtheory.org/SDT/documents/2004_DeciVansteenkiste_SDTandBasicNeedSatisfaction.pdf

Ernstmeyer, K., &amp; Christman, E. (2022). Therapeutic communication and the nurse-client relationship. In www.ncbi.nlm.nih.gov. Chippewa Valley Technical College. https://www.ncbi.nlm.nih.gov/books/NBK590036/ (Original work published 2025)

Faissner, M., &amp; Braun, E. (2023). The ethics of coercion in mental healthcare: The role of structural racism. Journal of Medical Ethics, 50(7). https://doi.org/10.1136/jme-2023-108984

Gilburt, H., Rose, D., &amp; Slade, M. (2008). The importance of relationships in mental health care: A qualitative study of service users’ experiences of psychiatric hospital admission in the UK. BMC Health Services Research, 8(1). https://doi.org/10.1186/1472-6963-8-92

Gooding, P., Mcsherry, B., Roper, C., &amp; Grey, F. (2018). Alternatives to coercion in mental health settings: A literature review. https://socialequity.unimelb.edu.au/__data/assets/pdf_file/0012/2898525/Alternatives-to-Coercion-Literature-Review-Melbourne-Social-Equity-Institute.pdf

Hamovitch, E. K., Choy-Brown, M., &amp; Stanhope, V. (2018). Person-Centered Care and the Therapeutic Alliance. Community Mental Health Journal, 54(7), 951–958. Springer Nature Link. https://doi.org/10.1007/s10597-018-0295-z

Hem, M. H., Gjerberg, E., Husum, T. L., &amp; Pedersen, R. (2018). Ethical challenges when using coercion in mental healthcare: A systematic literature review. Nursing Ethics, 25(1), 92–110. Sage Journals. https://doi.org/10.1177/0969733016629770

Iversen, H. W., Riley, H., Råbu, M., &amp; Lorem, G. F. (2025). Building and sustaining therapeutic relationships across treatment settings: A qualitative study of how patients navigate the group dynamics of mental healthcare. BMC Psychiatry, 25(1). https://doi.org/10.1186/s12888-025-06874-5

Johnson, L. N., &amp; Wright, D. W. (2002). Revisiting bordin’s theory on the therapeutic alliance: Implications for family therapy. Contemporary Family Therapy, 24(2), 257–269. https://doi.org/10.1023/a:1015395223978

Kornhaber, R., Walsh, K., Duff, J., &amp; Walker, K. (2016). Enhancing adult therapeutic interpersonal relationships in the acute health care setting: An integrative review. Journal of Multidisciplinary Healthcare, 9(14), 537–546. Pubmed Central. https://doi.org/10.2147/JMDH.S116957

Newton-Howes, G., &amp; Mullen, R. (2011). Coercion in psychiatric care: Systematic review of correlates and themes. Psychiatric Services, 62(5), 465–470. https://doi.org/10.1176/ps.62.5.pss6205_0465

Ng, J. Y. Y., Ntoumanis, N., Thøgersen-Ntoumani, C., Deci, E. L., Ryan, R. M., Duda, J. L., &amp; Williams, G. C. (2012). Self-Determination theory applied to health contexts. Perspectives on Psychological Science, 7(4), 325–340. https://doi.org/10.1177/1745691612447309

Norcross, J. C. (2010). The therapeutic relationship. Psycnet.apa.org; American Psychological Association. https://doi.org/10.1037/12075-004

Opland, C., &amp; Torrico, T. J. (2024, October 6). Psychotherapy and therapeutic relationship. National Library of Medicine; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK608012/

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68

Ryan, R. M., &amp; Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753

Sashidharan, S. P., Mezzina, R., &amp; Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350

Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version

Sharma, N., &amp; Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/

Stanhope, V., Marcus, S., &amp; Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183

Wostry, F., Hahn, S., &amp; Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012

}}

==External links==
* [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com)
* [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing)

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      <text bytes="13781" sha1="deahb514hhghrr6m9od9vdh60puz33m" xml:space="preserve">== interesting resource == 
Found this reference related to your topic and thought you might enjoy reading it too!
Sheehan, K. A., &amp; Burns, T. (2011). Perceived Coercion and the Therapeutic Relationship: A Neglected Association Psychiatric Services, 62(5), 471–476. https://doi.org/10.1176/ps.62.5.pss6205_0471

* The perceived coercion and therapeutic relationship are supposedly related in psychiatric admissions, and that therapeutic relationship quality is a neglected factor in how patients experience coercion. Even voluntary admission can feel coercive, depending on the interpersonal quality of care. 
* As nearly 48% of voluntarily admitted patients report high perceived coercion and among involuntarily admitted patients, it's ≈89%. 

That statistic was truly shocking!

--[[User:Christie M.B|Christie M.B]] ([[User talk:Christie M.B|discuss]] • [[Special:Contributions/Christie M.B|contribs]]) 00:20, 10 October 2025 (UTC)

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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:38, 19 August 2025 (UTC)

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&lt;!-- Written expression – Proofreading --&gt;
# Proofreading
## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard
&lt;!-- Written expression – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY]
&lt;!-- Written expression – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
&lt;!-- Written expression – Figures --&gt;
## Figures
### Very well captioned
&lt;!-- Written expression – Figures - Citations --&gt;
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Written expression – Tables --&gt;
## Tables
### Very well captioned
&lt;!-- Written expression – Tables - Citations --&gt;
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Written expression – Citations --&gt;
## Citations use very good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]:
### List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
### Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
&lt;!-- Written expression – References --&gt;
## References use good APA style:
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Check and correct use of italicisation
|8=
&lt;!-- Learning features comments... --&gt;
# Basic use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Basic use of figure(s)
# Very good use of table(s)
# Basic use of feature box(es)
# Basic use of scenarios, case studies, or examples
&lt;!-- Learning features – Quizzes --&gt;
# Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
&lt;!-- Learning features – See also --&gt;
# Basic use of the &quot;See also&quot; section
## Also include links to related book chapters
&lt;!-- Learning features – External links --&gt;
# Insufficient use of the &quot;External links&quot; section
## Link to the top 3-6 external resources about this topic
|9=
&lt;!-- Social contribution comments... --&gt;
# ~8 logged, useful, mostly minor/moderate/major contributions with direct links to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:00, 17 October 2025 (UTC)

== Images ==

I removed the images used in this lecture because they didn’t seem at all relevant. One is a random picture of a doctor on his computer and the other is an AI generated conversation between a therapist and patient; neither illustrates coercion in medical care. It’s like using a picture of an un-drank glass of orange juice to represent the topic of humans drinking— someone could hypothetically drink the orange juice, but it in isolation it has nothing to do with the action of drinking. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 23:25, 20 May 2026 (UTC)</text>
      <sha1>deahb514hhghrr6m9od9vdh60puz33m</sha1>
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  </page>
  <page>
    <title>Solar energy/Resource availability</title>
    <ns>0</ns>
    <id>324059</id>
    <revision>
      <id>2810670</id>
      <parentid>2760445</parentid>
      <timestamp>2026-05-20T22:38:12Z</timestamp>
      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
      </contributor>
      <minor />
      <comment>/* Hour angle, HA */</comment>
      <origin>2810670</origin>
      <model>wikitext</model>
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      <text bytes="9037" sha1="0ee2d8cn1m1y7nbxedmqntqx47cch6c" xml:space="preserve">== Introduction ==
The amount of sunlight that will strike the Earth's surface will depend on a few factors, some of which are periodic and others can be considered variable. Some periodic factors:

* Seasonal
* Day/night
* Urban shade

Some variable factors

* Cloud coverage
* Other natural weather events
In order to calculate the daily paths of the sun over the horizon across the year, several formulas and equations are first necessary to solve.

== Latitude and longitude ==
In addition to the position of the planet around the sun, the amount of sunlight available at any point on the Earth's surface will also depend on the longitude and latitude of the geographic position. Latitude is a measure of how far north or south you are, and varies from -'''90° (90° south) to +90° (90° north).''' The longitude is a measure of how far east or west one is, and varies from -180'''°''' '''(180° west) to +180° (180° east).'''

== True solar time ==
True solar time (TST) is used to convey the relative position of the sun compared to the earth, regardless of your position on earth. The following equation returns the true solar time in hours.

&lt;math&gt;TST = UTC + \frac{longitude}{15} + EoT&lt;/math&gt;

== Equation of time ==
The equation of time can be approximated by a sum of two sine waves. The first method is precise, and the second method shown here is approximate.

=== Precise method ===
&lt;math&gt;EoT = \Delta t_{ey} =  -7.659\sin(D) + 9.863\sin \left(2D + 3.5932 \right)&lt;/math&gt;(minutes)

where: &lt;math&gt;D = 6.240\, 040\, 77 + 0.017\, 201\, 97(365.25(y-2000) + d)&lt;/math&gt;

where &lt;math&gt;d&lt;/math&gt; represents the number of days since 1 January of the current year, &lt;math&gt;y&lt;/math&gt;.

=== Rough approximation ===
For &lt;math&gt;EoT&lt;/math&gt; in minutes, see the following equation. Ensure that the trigonometry relations are calculated in ''radians'' mode.

&lt;math&gt;EoT (minutes) = \Delta t_{ey} = 7.678\sin ({B + 1.374}) - 9.87\sin ({2B})&lt;/math&gt;

Where B is:

&lt;math&gt;B = 2 \pi \frac {(n-81)}{365}&lt;/math&gt;

Where &lt;math&gt;n&lt;/math&gt; is the day of the year.

For &lt;math&gt;EoT&lt;/math&gt; in degrees, see the following equation. Ensure that the trigonometry relations are calculated in ''degrees'' mode.

&lt;math&gt;EoT (degrees) = 0.0072cos(J) - 0.0528cos(2J) -0.0012cos(3J) - 0.1229sin(J) - 0.1565sin(2J) - 0.0041sin(3J)&lt;/math&gt;Where J is:

&lt;math&gt;J = n \times \frac {360}{365}&lt;/math&gt;

Where &lt;math&gt;n&lt;/math&gt; is the day of the year.

Note that some formulas result in positive answers, while other result in negative times. This is correct and refers to how early or late compared to true solar time the sun will be. For example, the sun can be as much as 14.29 minutes or 3.56 degrees slow (a positive amount) on [[w:_February_11|February 11th]] (the 42nd day of the year), and as much as 16.40 minutes or -4.10 degrees fast (a negative amount) on [[w:November_3|November 3rd]] (307th day of the year). 

== Solar declination, δ ==
Throughout the annual orbit of the Earth around the Sun, the distance between the two bodies is around 150 million km. Because the Earth's orbit is elliptical, the actual distance between the Sun and the Earth will vary somewhat. The Earth takes around 365.25 days to completely orbit the Sun. 

'''The solar declination, δ, is the angle at which the sun Earth is inclined relative the plane of orbit around the Sun. For the northern hemisphere, the angle of declination varies from +23.45 degrees in Winter to -23.45 degrees in Summer.'''

The solar declination angle can be calculated at any point of the year with the following equations:

=== Solar declination rough approximation, in degrees ===
&lt;math&gt;\delta_{degrees}= 23.45sin(360(n+284)/365)&lt;/math&gt;

=== Solar declination finer result, in radians ===
&lt;math&gt;\delta_{radians}= arcsin(0.369sin(360(n-82)/365 +2sin(360(n-2)/365))&lt;/math&gt;

=== Solar declination finer result, in degrees&lt;math&gt;\delta_{radians}= 0.33281 - 22.984cos(J) - 0.3499 cos(2J) - 0.1398cos(3J) + 3.7872sin(J) + 0.03205sin(2J) + 0.07187(3J0&lt;/math&gt; ===
Where J is the day of the year, with January 1st being n=1.

&lt;math&gt;J = n \times (360/365)&lt;/math&gt;, in degrees.

== Half hour angle, ω₀ ==
Calculate the hour angle when the altitude angle is equal to 0. 

&lt;math&gt;\omega_0 (\text {degrees})= arccos(-(\tan {\delta})* (\tan {Lat}))&lt;/math&gt;

Using this equation, you can find true solar time (TST) at sunrise and at sunset.

&lt;math&gt;TST_{sunrise} = 12 - (\frac {1}{15}) \omega_0&lt;/math&gt;, make sure that &lt;math&gt;\omega_0&lt;/math&gt;is in degrees.

&lt;math&gt;TST_{sunset} = 12 + (\frac {1}{15}) \omega_0&lt;/math&gt;, make sure that &lt;math&gt;\omega_0&lt;/math&gt;is in degrees.

In addition you can find day length, i.e. the time that the sun is above the horizon, using the half hour angle equation:

&lt;math&gt;S_0 = (\frac {2}{15}) \omega_0&lt;/math&gt;, make sure that &lt;math&gt;\omega_0&lt;/math&gt;is in degrees.

A quick check of the half hour angle and whether or not it is greater or less than &lt;math&gt;{90}^\circ &lt;/math&gt; can tell you if the day is shorter or longer than the night.

if: &lt;math&gt;\omega_0 &gt; 90&lt;/math&gt;, then the day is longer than the night, and the time that the sun is in the sky is at least 12 hours.

if: &lt;math&gt;\omega_0 &lt; 90&lt;/math&gt;, then the day is shorter than the night, and the time that the sun is in the sky is less than 12 hours.

== Elevation angle, h ==
Imagine the sun is up and you are standing outside. Turn your body so that you facing the sun. Then point your arm directly towards the sun. The angle between the ground (the plane surface, if you are standing on a hill ignore the angle of the hill) and your arm is the elevation angle. The function for the angle of elevation in degrees is:

&lt;math&gt;h = arcsin[cos(\delta - {Lat})] = 90^{\circ} - Lat + \delta&lt;/math&gt;

Where &lt;math&gt;\delta&lt;/math&gt; is the angle of declination in degrees, &lt;math&gt;Lat&lt;/math&gt; is the latitude of the location under analysis in degrees.

&lt;math&gt;h = arcsin(\cos {\delta}\cos{HA}\cos{Lat} +\sin{\delta}\sin{Lat})&lt;/math&gt;

== Hour angle, HA ==
The hour angle (noted HA) is the conversion of true solar time (TST) to an angle. Where solar midnight is always 0:00am or 24:00 - or +180 degrees and true solar noon is always 12:00 and 0 degrees. The calculation of the half hour angle, HA is:

For HA in degrees:

&lt;math&gt;HA = (TST - 12) \times \frac {360}{24}&lt;/math&gt;

And as there are 2 pi radian in 360 degrees of rotation (one rotation), the equation for HA in radians is:

&lt;math&gt;HA = (TST - 12) \times \frac {2 \pi}{24}&lt;/math&gt;

As can be  deduced from the equations, the sun moves 15 degrees (360/24) in the ecliptic plane, ''no matter what day of the year it is.''

This angle is measured by this type of sundial:

[[File:CICR monument NL 05.jpg|200x200px]]

== Azimuth angle, Az ==
The first step before calculating the azimuth angle is determining whether if the &lt;math&gt;sin(az)&lt;/math&gt; is positive or negative. 

The equation for &lt;math&gt;sin(az)&lt;/math&gt; is the following equation. Ensure that the trigonometry functions are set to ''degrees'' mode for calculating &lt;math&gt;\delta&lt;/math&gt;, &lt;math&gt;HA&lt;/math&gt;, and &lt;math&gt;h&lt;/math&gt;, if you converted the variables to ''degree'' form.

&lt;math&gt;sin(az) = \frac {cos(\delta) sin(HA)}{cos(h)}&lt;/math&gt;

Where &lt;math&gt;\delta&lt;/math&gt; is solar declination in ''degrees'', &lt;math&gt;HA&lt;/math&gt; is half hour angle in ''degrees'' and &lt;math&gt;h&lt;/math&gt; is altitude angle in ''degrees''.

if &lt;math&gt;sin(az) &lt; 0&lt;/math&gt; then use &lt;math&gt;az = - arccos (\frac{sin (Lat) sin(h) - sin(\delta)}{cos(Lat) cos(h)})&lt;/math&gt;

if &lt;math&gt;sin(az) &gt; 0&lt;/math&gt; then use &lt;math&gt;az = arccos (\frac{sin (Lat) sin(h) - sin(\delta)}{cos(Lat) cos(h)})&lt;/math&gt;

Where &lt;math&gt;Lat&lt;/math&gt; is the latitude in ''degrees'', &lt;math&gt;\delta&lt;/math&gt; is the solar declination angle in ''degrees'', and &lt;math&gt;h&lt;/math&gt; is the altitude angle in ''degrees''.

Note, that:

* if &lt;math&gt;tan(Lat) tan(\delta) &gt; 1&lt;/math&gt;, then the Sun is still up above the horizon (polar summer in the northern hemisphere).

* if &lt;math&gt;tan(Lat) tan(\delta) &lt; 1&lt;/math&gt;, then the Sun will not rise (polar winter in the northern hemisphere).

== Method for calculating sun path chart ==

=== For true solar noon of a given day of the year ===

# Find &lt;math&gt;n&lt;/math&gt; of year (day # of year, with January 1st being &lt;math&gt;n=1&lt;/math&gt;.)
# Calculate solar declination angle &lt;math&gt;\delta&lt;/math&gt; in degrees.
# Calculate equation of time (&lt;math&gt;EoT&lt;/math&gt;) in (minutes). Ensure that &lt;math&gt;EoT&lt;/math&gt; is converted to hours before the next step using the conversion ratio of 60 minutes per hour.
# For a select number of true solar times (&lt;math&gt;TST&lt;/math&gt;), for example, 8am, 9am, 10am, ... 10pm, etc. calculate the coordinated universal time (&lt;math&gt;UTC&lt;/math&gt;) and then the local time (&lt;math&gt;LT&lt;/math&gt;).
# Calculate the hour angle &lt;math&gt;HA&lt;/math&gt; for each of the true solar times under consideration.
# Calculate the altitude angle &lt;math&gt;h&lt;/math&gt; in degrees for each hour angle, &lt;math&gt;HA&lt;/math&gt;.
# Calculate the &lt;math&gt;sin(az)&lt;/math&gt; to determine if the value is negative or positive.
# Based on the relative positivity or negativity of the &lt;math&gt;sin(az)&lt;/math&gt;, choose the correct azimuth (&lt;math&gt;az&lt;/math&gt;) formula to use.

...

{{BookCat}}</text>
      <sha1>0ee2d8cn1m1y7nbxedmqntqx47cch6c</sha1>
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  </page>
  <page>
    <title>Introduction to solar energy</title>
    <ns>0</ns>
    <id>324869</id>
    <revision>
      <id>2810671</id>
      <parentid>2809337</parentid>
      <timestamp>2026-05-20T22:39:18Z</timestamp>
      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
      </contributor>
      <comment>/* Unorganized equations */</comment>
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      <text bytes="1295" sha1="lnbvxozf0qkfr5clgc91pauw0r2orxq" xml:space="preserve">This will be a course on solar energy. This course will cover solar energy and applications in solar photovoltaic, solar thermal and hybrid systems. This course is part of the [[solar energy]] topic.

Some of the learning objectives in this course are:

* Understanding solar resources and availability
* Understanding radiation as far as it relates to harvesting solar power
* Understanding optics as it relates to solar power collection and conversion

== Chapters ==

# [[Introduction to solar energy/Introduction|Introduction]]
# [[Introduction to solar energy/Solar resources and availability|Solar resources and availability]]

== Exercises ==

# [[Introduction to solar energy/Solar resources and availability quizbank 1|Solar resources and availability quizbank 1]]

== External resources ==

* [https://sites.engineering.ucsb.edu/~bennett/heatlib/rad/view/index.html View factor programs]

== Unorganized equations ==
[[Introduction to solar energy/Equations]]

Useful thermal heat output:

&lt;math&gt;\dot{q''_u} = \frac {\dot {m_f} \cdot C_{p,f} \cdot (T_{f,leaving} - T_{f,entering})}{A}&lt;/math&gt;

Fill Factor is the ratio of the maximum power to the product of open-circuit voltage and short-circuit current:

&lt;math&gt;FF=\frac {P_{max}}{V_{oc}\cdot {I_{sc}}}&lt;/math&gt;
[[Category:Solar energy]]</text>
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  </page>
  <page>
    <title>Wikiversity:Colloquium/archives/October 2025</title>
    <ns>4</ns>
    <id>325670</id>
    <revision>
      <id>2810761</id>
      <parentid>2791625</parentid>
      <timestamp>2026-05-21T10:43:14Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Curators and curators policy */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2810761</origin>
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      <text bytes="54279" sha1="9g7ft6k87jpatoxtaqh36h6oy2szx9s" xml:space="preserve">== Publishing text from ResearchGate in Wikiversity as a copy ==
I am dealing with someone who wants to place a copy of an article he (and his colleagues) published on ResearchGate. He is now in the process of attaching a license compatible with Wikiversity. The article on ResearchGate is fully publicly accessible, and Google indexes its word sequences (unless broken by a newline).

Do we want to allow that kind of publishing in Wikiversity?

All I see in it is a ''duplicate'', no added value. The person seems to think adding the article to Wikiversity is going to help LLMs and findability, but I do not see why it should be the case. The article is accessible as a web page; a pdf is available from a link.

I sense that if we allow that, we can expect a surge of copies from CC-BY-SA 4.0 licensed material from ResearchGate, with no or little appreciable benefit for anyone.

I do not know whether ResearchGate's conditions prohibit republishing. I guess that if they allow licensing as CC-BY-SA, they cannot prohibit copies from being made.

Thank you, and sorry for the bother. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:45, 3 October 2025 (UTC)

:[[User talk:Dan Polansky#Clarification on authorship and licensing for Enhancing Web Browser Security through Cookie Encryption|The Entire discussion can be found here]]
:[[File:A_screenshot_of_the_double-column_layout.png|thumb|A Double Column Layout]]
: For the record, I have already explained in earlier discussions with Dan why this article deserves to stay on Wikiversity. Instead d addressing those points, he deleted it outright and dismissed my reply as “GenAI slop.”

: Dan stated:''&quot;The article in ResearchGate is fully publicly accessible and Google finds its word sequences (unless broken by a newline).&quot;''

: No, this isn't true. Reasons: 1) Google indexed that page because of the content description, which we had entered when uploading the document. 2) This Document IS NOT ACCESSIBLE on mobile unless we download and use a separate app to open it.

: Creating a Wikiversity article would ensure that it is accessible from any device, and I can convert the double-column layout into a single-column layout since that would not confuse the LLMs that are training on public data.

: The current double-column layout is problematic as it is not being read correctly by the machines.

: Besides, my colleague was in the process of optimizing the content for the Wikiversity audience by making changes to the content and adding internal links &amp; references. But before he could do that, Dan deleted the page without prior warnings or any discussions, thinking that it was a copyvio.

[[File:A_mobile_screenshot_of_Research_Gate_page_of_an_article.jpg|thumb|A proof that the document isn't visible from the mobile]]

:I  strongly believe that this article would be useful for students and those who are in the IT sector.

: Since this issue began, I've been reading the Wikiversity rules again and again. I would like to remind you that
 
:a) Wikiversity already hosts ''Wikipedia mirrors'' (which are available publicly elsewhere), proving that “duplicate existence” is not disqualifying.

:b)Wikiversity encourages ''educational hosting'' of materials, unique research. If ResearchGate has a PDF, it doesn’t mean Wikiversity can’t host a textual, open-licensed educational version; therefore, I don’t think it is fair for one individual to dismiss it as “no added value.” Wikiversity is a collaborative space, and whether something is valuable should be determined by community consensus, not unilateral action.

:c) I am one of the authors, nd I do have the documents to prove it

:d) I did mention the license in my UserTalk, ResearchGate, and Zenodo. Please let me know if you require any further clarification.  
:[[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 02:25, 5 October 2025 (UTC)
::'''TL;DR:'''
::The ResearchGate version is not fully accessible (especially on mobile) and features a double-column layout, which is poor for readability and machine indexing.
::- Wikiversity allows duplicates (e.g., Wikipedia mirrors) — so “already exists online” is not grounds for deletion.
::- I was improving the article with links/references for Wikiversity’s educational use when it was deleted prematurely.
::- This is a community decision, not one person’s. I request that custodians review this matter. 
:: For the record, I have edited a few lines and rephrased them., You can refer to them in this section of the history.[[User: Tomlovesfar|Tomlovesfar]] ([[User talk :Tomlovesfar|discuss]] • [[Specia l:Contributions/Tomlovesfar|contribs]]) 02:25, 5 October 2025 (UTC)
::: To prevent the discussion from being derailed, I addressed problems with the above on the tak page of Tomlovesfar. I ask other editors, especially custodians and curators, for input. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 02:18, 5 October 2025 (UTC)
::::I will be recreating the article ''“Enhancing Web Browser Security through Cookie Encryption”'' once again.
::::Authorship and licensing have already been proved in the discussion, The work is released under CC BY-SA 4.0, with the authors’ consent clearly stated on both ResearchGate and Zenodo.
::::If anyone still has concerns or questions, please feel free to discuss them on the article’s '''talk page''' so that the discussion remains focused and traceable.
::::My goal is to ensure collaborative improvement rather than dispute. [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:15, 6 October 2025 (UTC)
:So it's not about the text as such, but about the purpose. It shouldn't be judged by how much text, copy, or the like, but by what the purpose is. If the purpose is educational or research, then hosting the text shouldn't be a problem. If there's another purpose, then definitely not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 13:07, 16 October 2025 (UTC)

== Looking for interested collaborators in engineering ==

Hi all, I know this is not quite the right place for this comment, but I'd figure I post a note here, in case there are any other people interested in collaborating on any engineering topics. I've been working on a thermodynamics course, that I could use some help with. Also, in general the entire school of engineering could use some work to help organize the sub-portals and topics. Feel free to drop a message on my user page! [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:32, 4 October 2025 (UTC)

== Quantum: A Walk Through the Universe ==

Hi and Hello. Beeing new to Wikiversity I used sandbox to write an article, ofcourse without reading the manual. Then moved the page to  Wikiversity. Just found out how to do it properly, but i have now 2 pages: [[Quantum: A Walk Through the Universe]] and [[Wikiversity:Quantum: A Walk Through the Universe]]. Is there anyone that could remove the unwanted page? Thanks in advance. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:51, 5 October 2025 (UTC)
: I made changes to your post that should be acceptable: removed a strange tick and added wikilinks to the items you are talking about, for ease of administration. I can delete  [[Wikiversity:Quantum: A Walk Through the Universe]], which would make more sense given the prefix &quot;Wikiversity&quot;, but I can also delete the other one and then move it; in the end, the prefix does not matter and any of the two can be kept as the main one. I guess one would perhaps want to keep the item with larger editing history, which would be [[Wikiversity:Quantum: A Walk Through the Universe]]?
: Disclaimer: I make no represenation at this time, not even implied, about whether this should be kept or whether this is pseudo-physics (of which we get a bit too much). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:05, 5 October 2025 (UTC)
::@[[User:Dan Polansky|Dan Polansky]] Thanks for you kind help. I would like the Wikiversity:Quantum: A Walk Through the Universe to be deleted, since it does not follow wiki rules (I think). Any tips welcome !!! In Wikipedia we have a &quot;Teahouse&quot; I guess this is simmilar. Also in WP there is a &quot;submit for review&quot; button, thats what i could not find here. I gladly would put it up for review. Thanks again. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:47, 5 October 2025 (UTC)
::Just looked at the work you did... Thank you !! I was in the process of correcting them, you saved me time and effort. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:59, 5 October 2025 (UTC)
::: I went ahead and deleted the page with very little editing history and then moved the page with more editing history to the right place. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:06, 5 October 2025 (UTC)
::::Great ! Exactly what i wanted. Did you see the Quantum Cheat Sheet ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:31, 5 October 2025 (UTC)

== Creation of a department of plurilingual education ==

Hi, 
I'm the coordinator of an EU-project (PEP - Promoting plurilingual education, cofounded by the European commission in the Erasmus+ programme). We are developing lessons (almost 40) on plurilingual education in French and English. We already have a [https://fr.wikiversity.org/wiki/D%C3%A9partement:Didactiques_du_plurilinguisme department in the French Wikiversity] (faculty of Education), which we have started to feed. 
We would like to open a similar department in the English Wikiversity. Can you help us? Thanks in advance. 
Best wishes 
Christian Ollivier [https://fr.wikiversity.org/wiki/Utilisateur:Projet_PEP Projet PEP|Projet PEP] 10:44, 6 October 2025 (UTC)
: What kind of help would you like to see? Like, do you have some specific questions you would like to ask, concerning what is possible in the English Wikiversity?
: A transparent link to your English Wikiversity user account: [[User:Projet PEP]]. As a first step, I would recommend you create a page at [[Meta:User:Projet PEP]]: this will then get automatically shown across wikis in various languages. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:24, 6 October 2025 (UTC)

:Hi Dan,
:Thanks for your reply. We would need help to create a department of plurilingual education in the facutlty of education or in the school of language studies. 
:We already had an account in the French version, I have now created one in the English version of Wikiversity. 
:Thanks in advance for your support.
:Christian @[[User:Projet PEP|Projet PEP]] [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) 07:19, 15 October 2025 (UTC)
:: I do not know what is is to ''create a department of education''. That may be my fault since I do not care about ''departments''; I care about expository articles. Searching for &quot;WV:Department&quot;, I find no unequivocal evidence that the English Wikiversity has something like a ''department'' concept. I found [[:Category:Department of Occupational and Environmental Health Sciences, University of Panamá]], but that looks suspect to me.
:: One thing you can do is create something in your user space, e.g. [[User:Projet PEP/Department of plurilingual education]]; that would not require any pre-approval or the like, I think.
:: Again, I am perhaps not the best person to ask; I am providing information on your query that I could quickly figure out. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:45, 15 October 2025 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:15, 9 May 2026 (UTC)}}
How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
{{archive bottom}}

== Physics - Any Educators willing to help with [[Quantum]] ? ==

While writing the page [[Quantum]] I am not satisfied with the flow, sound and looks of the page.
Objectives:
* Educational
* Attractive to students of any age
* More colorfull
If you look at the page as is, its boring :)
Any help welcome.. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 15:34, 19 October 2025 (UTC)

:I will help you with the page. Regards —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 17:46, 19 October 2025 (UTC)
::Thank you!! If you look at [[W:Quantum]] that  is exactly what i don't want :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 20:52, 19 October 2025 (UTC)
:::@[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] Hi, I changed the page a bit, do you like it?  [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 13:35, 21 October 2025 (UTC)
::::haven't checked yet, but i'm sure it's good! —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 16:26, 21 October 2025 (UTC)

== Concern regarding curator conduct User:Dan Polansky ==

I’d like to raise a concern about the recent interactions I had with curator [[User:Dan Polansky|Dan Polansky]] during a discussion about one of my research articles.

My issue is not about the deletion of my article itself, but about his repeated unprofessional conduct and dismissive tone throughout the process.

Dan deleted my article assuming it was a copyright violation, even though it had a CC BY 4.0 license clearly shown on [https://zenodo.org/records/15287972 Zenodo]. I even updated the ResearchGate description to confirm that the authors (myself included) consented to publication on Wikiversity. Instead of moving toward resolution, the discussion kept looping first copyright, then my identity, then grammar, and then back to copyright again.

[[User talk:Dan Polansky#Clarification on authorship and licensing for Enhancing Web Browser Security through Cookie Encryption|He also removed one of my replies from his talk page]], calling it “GenAI slop,” even though that message contained proof of my authorship. That felt dismissive and unnecessary, especially when I was trying to clarify things in good faith.

At one point, he admitted he was inexperienced in handling such matters and told me to ask other curators to step in, which is fine... but if that’s the case, deleting an entire page without prior discussion seems premature. Even after I proved authorship and clarified the license, he never reverted the deletion.

Later, he went on to write a public note in his own sandbox describing our interaction, saying he “[[User:Dan Polansky/Blog#Wikipedia, Wikiversity and GenAI slop|at the beginning of Oct 2025, I interacted with someone who threw at me GenAI slop]]” directly referencing me. That felt quite unprofessional and disrespectful, especially coming from a curator.

Finally, he began micromanaging my sandbox, criticizing how I structured my own draft and telling me to “[[User talk:Tomlovesfar#User:Tomlovesfar/sandbox|move authors to the top,” while also pointing out that “some sections were missing]]” compared to the ResearchGate version. As an author, I have the right to adapt or shorten my own work that’s what a sandbox is for. When I reminded him of that, he responded by calling me “[[User talk:Tomlovesfar#User:Tomlovesfar/sandbox|a major waste of time and attention.]]”

While I understand moderators have to check for possible copyright issues, this tone and behavior felt personally belittling and not in the spirit of collaboration or respect that Wikiversity encourages.

I’d appreciate if the custodians could review this matter and consider whether this kind of conduct is appropriate for someone in a curator position.

I would also like to kindly request that Dan refrain from responding to this thread until at least one other curator or custodian has shared their view, so the discussion can remain balanced and impartial.

''[[wikipedia:Wikipedia:Please_do_not_bite_the_newcomers|WP:DNB]]  |   [[wikipedia:Wikipedia:Snowball_clause|WP:SNOW]]  |   [[wikipedia:Wikipedia:Assume_good_faith|WP:FAITH]]'' [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 18:56, 6 October 2025 (UTC)
: I will happily cease interaction with user Tomlovesfar and cease handling of his content, handing it over to any curator or custodian who volunteers to replace my role. Then, I guess I could ignore his activities going forward and pass the responsibility to others. So far, I have acted to protect the integrity of the English Wikiversity. I posted problems I identified to [[User talk:Tomlovesfar]], to make it easy to understand to others. It suffices you (curators and custodians) notify me that I should disengage and I will. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:08, 6 October 2025 (UTC)
::Is there a policy on en.wv to prohibit AI assisted communication? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:19, 6 October 2025 (UTC)
:::I have a similar question: Is there a polich on en.wv to prohibit text generated by LLMs? I've just (today) launched a project ([[AIXoer]]), and will soon launch another, that teaches students how to collaborative use LLMs to generate text for wikiversity pages. Students will build LLM-based hypertexts in wikiversity. At the same time, they will complete the Wiki.edu assignments to edit / improve wikipedia articles on hypertext, artificial intelligence, etc.....
:::I figure it's ok until or unless I hear otherwise. [[User:Stevesuny|Stevesuny]] ([[User talk:Stevesuny|discuss]] • [[Special:Contributions/Stevesuny|contribs]]) 21:29, 6 January 2026 (UTC)
::::{{ping|Stevesuny}} Welcome back to Wikiversity. The relevant discussion is at [[Wikiversity:Artificial intelligence]]. I plan on giving my input soon and I encourage everyone to do the same. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:09, 6 January 2026 (UTC)
:::::@[[User:Atcovi|Atcovi]] Would  it be wise to open a new topic? I dont see the connection with this topic. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 22:58, 6 January 2026 (UTC)
::::::{{ping|Harold Foppele}} No problem with opening a new topic on the matter. I was just answering Stevesuny's question, but I think we can continue any discussions relating to the usage of LLMs on Wikiversity here: [[Wikiversity talk:Artificial intelligence]] (or maybe someone can open a community review?). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 23:55, 6 January 2026 (UTC)
::BTW, you stayted &quot;[https://en.wikiversity.org/w/index.php?title=Special:Log&amp;logid=3456363 ...and I do not see that the researchgate license would be Wikiversity compatible]&quot;. As you can see f[https://www.researchgate.net/publication/391195563_Securing_and_Enhancing_Web_Browser_Security_through_Cookie_Encryption rom the link], the license is compatible. The question is if the reasource was created the proper way mentioning all authors. (posibly original source). [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:34, 6 October 2025 (UTC)
::: I was asked above by Tomlovesfar to refrain from responding. I provided only a minimum response, offering to step away and yield to curators and custodians. I am refraining from further communication here not to derail the communication, awaiting input from curators or custodians. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 20:52, 6 October 2025 (UTC)
::::The person in question can request this, but it does not comply with the rules of the Wikimedia movement. Unfortunately, the person in question did not post it on the Request custodian action page, but in the discussion area, so I think it is normal to discuss it here. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:58, 6 October 2025 (UTC)
:::::@[[User:Tomlovesfar|Tomlovesfar]], @[[User:Juandev|Juandev]] i second the concern [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:17, 12 October 2025 (UTC)
:::::Since there hasn't been any reply from other curators, shall i proceed and post it on request custodian action page?  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 14:15, 12 October 2025 (UTC)
::::::@[[User:Tomlovesfar|Tomlovesfar]] Is your problem solved?[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • ]][[Special:Contributions/Harold Foppele|contribs]]) 09:17, 16 October 2025 (UTC)
::::::@[[User:Tomlovesfar|Tomlovesfar]] I had a look fast on policies and havent found a guideline for undeletion. So yes, I woudl directy ask custidians to undelete it via [[Wikiversity:Request custodian action]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:17, 16 October 2025 (UTC)
:::::::I was not talking about the article page, as I've already created it once again. I meant raising a post on request custodian action page regarding the behaviour of Dan. [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 19:49, 16 October 2025 (UTC)
::::::::The [[Wikiversity:Curators|unapproved policy on Curators]] states that one should first talk to the Curator and then to its Mentor if it doesn't work, which in this case is [[User:Jtneill]]. The goal is to find a consensus on the action. If it still fails, open the topic here on Colloquium to discuss with a broader community. Custodians will probably not help you at the moment, as there are no policies on this. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:17, 16 October 2025 (UTC)
:::::::::I'm assuming that he is ignoring the discussion, and it seems like no other custodian wants to join in the discussion too.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 03:51, 17 October 2025 (UTC)
: A small explanatory note about how I proceed as for a ''possible copyright violation''. I now deleted [[Zagatala State Nature Reserve]], as a result of searching for word sequences online and finding text containing those; then I tried to figure out whether the target would have copied from Wikiversity, determined that probably not, and proceeded to deletion. With the discussed item by Tom, I proceeded similarly, but only after I figured out it actually could have been copyright violation, and then went searching for text snippets and found some in ResearchGate, and ''at that time point'', there was no license attached to the article in ResearchGate. It did not cross my mind to visit Zenodo; I did not have that entity active in my mind at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:46, 9 October 2025 (UTC)
::BTW, you said, you are not going to comment on this and now you are comenting. So will you be discussing with us or no? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:18, 16 October 2025 (UTC)

My view: [[Wikiversity:Staff]] should be welcoming and supportive as well as doing administrative tasks. Editors, similarly, should be polite and respectful. We have a couple of newer contributors ([[User:Harold Foppele]] and [[User:Tomlovesfar]]) reporting similar issues (micromanaging, respect/politeness) with [[User:Dan Polansky|Dan Polansky]], a curator.
* Please take this feedback on board, [[User:Dan Polansky|Dan Polansky]]. Let's [[Wikiversity:Assume good faith]] and seek to educate/support in the first instance, especially with newer editors. That's how we grow a collaborative, healthy editing community. Only non-controversial deletions should be done without discussion and consensus.
* All edits need to be compliant with copyright. If appropriate to use generative AI, provide acknowledgement with link to the conversation/generation.
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:58, 17 October 2025 (UTC)

:Adding on to Jtneill's comment, I felt that Dan's language was very uncivil, and by looking into actions conducted by him in other Wikimedia projects, this is not new from him. Although some of his points were correct, I didn't really like him editing my user page, and I felt that his behaviour mirrored his behaviour on Wiktionary. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 05:25, 17 October 2025 (UTC)
::A note: Curators are supposed to follow '''all''' policies, including [[Wikiversity:Civility]] and [[Wikiversity:Assume good faith]]. This can be for a brief period, because some of this behaviour has carried on from the English Wiktionary. If you choose to test newbies, rant about other users, test newbies on their qualifications etc. I highly doubt it should be allowed. We need to feel welcome on Wikiversity, and not be discouraged from editing. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 05:37, 17 October 2025 (UTC)
::: Believing the above feedback, I guess I still have a lot to learn and adapt. Can you quote here the language that you found ''very uncivil'', so we can know what you are talking about specifically? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:02, 17 October 2025 (UTC)
::::Dan, I understand that you want to keep the wikiversity clean and useful for everyone. I agree with most of your points mentioned in [[User:Dan Polansky/Problem reports (about Wikiversity problems)|User:Dan Polansky/Problem reports (about Wikiversity problems) - Wikiversity]].
::::However, before making any changes to the page at least discuss it with the author. if the author does not reply within 48 hours, you may proceed with whatever you are meant to do. I have read your history and all the drama in Wiktionary, and I do not want you to get blocked here too. Therefore, please take this feedback and learn from your mistakes.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 07:10, 17 October 2025 (UTC)
::::: Just like Popper's beetles, I am trying to learn from mistakes, especially mine. As per Popper, when amoeba makes a serious mistake, it is eliminated; Einstein lets his theories die in his stead. How back to the point again. Now really, can you quote here the language that you found ''very uncivil''? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:14, 17 October 2025 (UTC)
::::::Have you noticed my notes in your talk page? For example: [[User talk:Dan Polansky#Testing newbies is not civil|User talk:Dan Polansky#Testing newbies is not civil]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:42, 17 October 2025 (UTC)
::::::: It should not be too hard for anyone to provide ''here'' a ''quote'' of actual language that is ''very uncivil''. Not too much work, right? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:44, 17 October 2025 (UTC)
::::::::Isn't calling the user &quot;A major waste of time&quot;
::::::::Uncivil? 
::::::::&lt;nowiki&gt;https://en.wikiversity.org/wiki/User_talk:Tomlovesfar&lt;/nowiki&gt; [[Special:Contributions/&amp;#126;2025-29083-30|&amp;#126;2025-29083-30]] ([[User talk:&amp;#126;2025-29083-30|talk]]) 07:54, 17 October 2025 (UTC)
:::::::::To pillory someone is uncivil. https://en.wikiversity.org/wiki/User:Dan_Polansky/Problem_reports_(about_Wikiversity_problems)
:::::::::To call a person &quot;This kind of people&quot; is scandalous.
:::::::::Do you mean like &quot;Negroes&quot;, &quot;Jews&quot;, &quot;Homosexuals&quot;, &quot;Americans&quot;? https://en.wikiversity.org/wiki/User_talk:Jtneill#c-Dan_Polansky-20251015041700-Jtneill-20251015040300
:::::::::Responding with DFX to a discussion is uncivil. https://en.wikiversity.org/wiki/User_talk:Jtneill#c-Dan_Polansky-20251015082600-Harold_Foppele-20251015082100
:::::::::Follow your mentor's lead, a simple guidance instead of focusing and escalating a problem.
:::::::::If you dont want to listen to critism nor helping people then you should not be a curator. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:59, 17 October 2025 (UTC)
::::::::I think it would be reasonable to expect you to check your own talk page. Just a note — your phrasing ''&quot;Not too much work, right?&quot;'' sounds slightly sharp.&quot; [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:50, 17 October 2025 (UTC)
::::::(Note) this comment was from User:Tomlovesfar, you are confusing them with me (User:RailwayEnthusiast2025) —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 15:05, 17 October 2025 (UTC)
:::::::Don't worry, we all understand what this is about :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 15:09, 17 October 2025 (UTC)
::::::::&lt;s&gt;Anyway, I believe @[[User:Dan Polansky|Dan Polansky]]'s conduct was really ruthless and mean. He was not very understanding and gave off the tone that he only believed that his view was right, as seen with the quotes above, as well as this link: [[User talk:RailwayEnthusiast2025/Basic Scratch Coding]]. Obviously, the page wasn't ready yet, and now I can see why it should be in userspace, but his tone gave off the fact that my work wasn't good enough and that I didn't have a voice, otherwise I would be silenced by [[Wikiversity:Request custodian action]] similar to the request made to @[[User:KayYayPark|KayYayPark]]. I also put a message on his talk page.  In that case, I moved the pages to my userspace, hoping that the drama would stop. It also seemed like he didn't read ''all'' of my text, as he only read the first comment and made his reply. I also made an assertive comment on his talk page, titled (Regarding Wikiversity:Colloquium) of which he didn't give any advice, and he ignored the fact that I was inexperienced. From now on, I'd like to cease contact with @[[User:Dan Polansky|Dan Polansky]], and instead of being too harsh, give clear reasons and follow policies. As I said, give advice instead of being too harsh; I have life outside of Wikiversity.&lt;/s&gt; There also isn't a policy on what ''is'' and ''isn't'' good enough for mainspace. I think there should be. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 15:36, 17 October 2025 (UTC)
:::::::::Looking at all the edits and reading all the comments I suggest to put @[[User:Dan Polansky|Dan Polansky]] on hold until there is a concensus regarding his position. [[Special:Contributions/&amp;#126;2025-29155-36|&amp;#126;2025-29155-36]] ([[User talk:&amp;#126;2025-29155-36|talk]]) 17:03, 17 October 2025 (UTC)
::::::::::Well, maybe if someone with authority at Wikiversity could talk some sense into @[[User:Dan Polansky|Dan Polansky]] ..... We should go by AGF ! [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:44, 17 October 2025 (UTC)
:::::::::::If you want to get community review, you can ask at [[Wikiversity:Community Review]], fyi. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 12:14, 25 October 2025 (UTC)
::::::::::::The discussion going on regarding Dan's behaviour at [[Wikiversity:Community Review/Dan Polansky]] resulted in &quot;temporarily removing the Curator flag&quot; by [[User:Mu301]]. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:02, 16 November 2025 (UTC)

== Contesting article deletions ==
What insitution or peron(s) does Wikiversity have to contest article deletion?--[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:29, 9 October 2025 (UTC)

:This: [[Wikiversity:Requests for Deletion]]. Its both for deletion and undeletion. This procedure is described in [[Wikiversity:Deletions|Deletions]] guideline. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 03:45, 17 October 2025 (UTC)

== Have your say: vote for the 2025 Board of Trustees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2025|2025 Board of Trustees election]] is now open. Candidates are running for two (2) seats on the Board.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia Foundation elections/2025/Voter eligibility guidelines|voter eligibility page]].

Learn more about them by [[m:Special:MyLanguage/Wikimedia Foundation elections/2025/Candidates|reading their application statements and watch their candidacy videos]].

When you are ready, go to the [[m:Special:SecurePoll/vote/405|SecurePoll voting page to vote]].

'''The vote is open from October 8 at 00:00 UTC to October 22 at 23:59 UTC.'''

Best regards,

Abhishek Suryawanshi&lt;br /&gt;Chair, Elections Committee&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 04:49, 9 October 2025 (UTC)
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== WMF Board reform ==

Wikiversity editors may be interested in the [[m:2025 WMF Board reform petition]]. [[User:Clovermoss|Clovermoss]] ([[User talk:Clovermoss|discuss]] • [[Special:Contributions/Clovermoss|contribs]]) 04:02, 11 October 2025 (UTC)

== Removal Request ==

Goodevening, could someone please remove all my userspace pages? I already marked them for deletion. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:07, 20 October 2025 (UTC)

:Hello @[[User:Harold Foppele|Harold Foppele]], I would recommend asking for this at [[Wikiversity:Request custodian action]], rather than in the main discussion. Because custodians will typically respond to it quicker over there afaik. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:52, 20 October 2025 (UTC)
::Will do, thx! [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:04, 20 October 2025 (UTC)

== a portal for learning step by step ==

hello there are apps like Moodle which allow to learn interactively step by step and for a student to record their progress. is there some app installed like this for Wikiversity. as i would like to make a course about coding gadgets for wiki, coding extensions for wiki, and how to make a page. and these contents put just on a wiki page would be difficult to progress. if there was a platform similar to udemy that exposes the wikiversity content to others i think it would be helpful. please let me know if this was already installed, if not, whether you would be interested to try it. thank you [[User:Gryllida|Gryllida]] 22:23, 16 October 2025 (UTC)

:This feature isn't available on Wikiversity, I think mainly because there is no student record (other than their editing history). For example, there is a [[Help:Quiz|quiz]] extension installed, but it does not store a record of student performance/progress. It is possible to embed Wikiversity content in learning management systems such as Moodle (e.g., using iFrame) around which a recorded step-by-step learning experience could be constructed. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:00, 17 October 2025 (UTC)
:It's as Jtneill writes. The reason why such an extension was not sought is probably that Wikiversity is not supposed to offer courses. This is not always followed, but the initial instruction from the Wikimedia Foundation was probably that Wikiversity was not supposed to award certificates. In general, there is also a problem that the development of the MediaWiki software is mainly done for Wikipedia, which makes other sister projects that would need different development and different tools suffer a bit. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 04:00, 17 October 2025 (UTC)
::Hi @[[User:Jtneill|Jtneill]] &amp; @[[User:Juandev|Juandev]], i think it would be good if 
::* course was easier to write (i wrote links to previous and next page on it manually as wikilinks)
::* users could bookmark their progress and optionally show it on their user page
::* User can add courses to &quot;i am doing this now&quot; and &quot;i wish I could do this one day&quot; lists
::* A user is listed as teacher and can take homework or verbal exam and sign a section on user page of a student
::Is there some code that kind of already does this. Would all seem doable to me except I got mad for the first item, it was exhausting to have a &quot;wizard&quot; format as there is no structure to present a next button or a previous button as link without loads of manual work...
::Please let me know what you think as I think it would make a lot of Wikiversity materials more accessible and easier to use. It could also help training new Wikipedia users in creating, editing and reviewing pages...
::Regards, [[User:Gryllida|Gryllida]] 10:00, 17 October 2025 (UTC)
:::These are great wishes and ideas, but as [[User:Juandev|Juandev]] explained Wikiversity is &quot;just&quot; an implementation of MediaWiki software which is software primarily tailored for Wikipedia. Wikiversity will not likely become even a light [[w:learning management system|learning management system]] and much more likely continue to allow for development of open educational resources (OERs). There are some other learning-oriented implementations of MediaWiki that might be interesting e.g., [https://wikieducator.org/ WikiEducator], but still, they focus on OERs. So, there a lot of freedom to structure and organise materials, but not really much prospect for storing/tracking learner progress. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:23, 17 October 2025 (UTC)
::::Would it not be practical to have a learning progress tracker thing as theen the courses would be easier to use. And with more users there would be more contributions to the courses themselves as well as to whatever was being trained on. Think like course &quot;how to start editing wiki&quot; or &quot;how to volunteer as an Ubuntu maintainer&quot;. They would create more users who know how to do these things. Seems in scope for Wikimedia mission. This could work by syncing Wikiversity content in a Moodle. [[User:Gryllida|Gryllida]] 11:36, 22 October 2025 (UTC)
:::::So you can either advocate with the mediawiki developers to develop something like this, or write it yourself and then propose it for inclusion in the mediawiki software, for example in the form of an extension. Some things could probably be solved with simple JavaScript, but the tracking would probably have to communicate with the server, so it would require a completely new extension. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:38, 26 October 2025 (UTC)

== Add the following user rights to the curator toolset? ==

Currently, curators can '''only''' delete pages (but not undelete them), but I propose that we add these permissions which will allow them to search for deleted pages, view deleted history entries, and to view deleted text (as curators are on par with custodians, regarding their trust and responsibilities):
* &lt;code&gt;browsearchive&lt;/code&gt; ({{int:right-browsearchive}})
* &lt;code&gt;deletedhistory&lt;/code&gt; ({{int:right-deletedhistory}})
* &lt;code&gt;deletedtext&lt;/code&gt; ({{int:right-deletedtext}})
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:29, 21 October 2025 (UTC)

:I'm not at all thrilled about this spontaneous emergence of a new wiki position. Not that I have anything against curators, but how is it possible that Wikiversity has curators but doesn't have an [[Wikiversity:Curators|approved policy about curators]]? In this case, I would suggest finalizing the policy about curators first and only then proposing some extensions of rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:53, 26 October 2025 (UTC)
:: A page with a policy status relating to curators is here: [[Wikiversity:User access levels]]; the vote is on the talk page. Therefore, the set up of curators and their rights/privileges was done via formally verified consensus. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:29, 7 November 2025 (UTC)

: From what I understand, the key benefit of the curator role is that it does not require as much trust as the custodians. And thus, curators have ''content''-related tools/powers whereas custodians have ''user''-related tools/powers in addition. A deleted page can contain user-identifying information, not just a content matter. And thus, using this logic/philosophy, curators should not be able to see deleted content (they are not trusted with user-relating items). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:45, 3 November 2025 (UTC)
:: Would this count as an objection? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:29, 7 November 2025 (UTC)
::: Yes, I think the above counts as an objection/con to your proposal. Also, let me quote multiple people from a ticket: &quot;WMF Legal has made it clear at https://en.wikipedia.org/wiki/Wikipedia:Viewing_deleted_content that they do not want sub-administrator user groups to have access to deleted content.&quot; and &quot;I agree with Geoff’s and Mike’s previous statements. It’s extremely important for the health and protection of the projects to limit access to deleted material that may include sensitive content. For this reason, we can’t support a proposal to create an assistant admin class that has access to deleted content before going through any kind of community selection process.&quot;; from [https://phabricator.wikimedia.org/T113109 T113109 New Assistant usergroup for en.wikiversity, two new rights for Custodians], phabricator.wikimedia.org; you can look at the ticket for context and other items of discussion. On the other hand, from the ticket: &quot;It looks like the WMF will still be happy with this request if (a) assistants go through a community selection process, or (b) they don't have access to deleted content.&quot; Don't take my quick interpretation and quotation selection to be solid and representative of the ticket; do your own reading. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:42, 7 November 2025 (UTC)
: I'll go ahead and withdraw this proposal. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:17, 2 December 2025 (UTC)

== Adding powerpoint ==

I’m thinking of adding a presentation (powerpoint in latex) that’s related to a wikiversity course I’m working on. Is PanDocElectron/Wiki2Reveal Demo a good alternative that allows others to work on the powerpoint in the future? Or can I create a page for the latex code? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 09:20, 22 October 2025 (UTC)

:Maybe just write a guideline, how to install ''PanDocElectron/Wiki2Reveal Demo'' and use it? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 26 October 2025 (UTC)
::Hey Juandev, thanks :). I’m not sure I follow you. I want to make a PowerPoint on psychrometrics, and be able to share it in a format that others can amend/work on/use without proprietary software. Are you suggesting that I just use the pandoc electron wiki2reveal demo? If that is the best option I can give that a shot, I didn’t know if other users or projects had uploaded/shared/collaborated on PowerPoints on wikiversity before. Thanks! [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:40, 30 October 2025 (UTC)
::: I don't know about PanDocElectron/Wiki2Reveal. But one option you have is to create a LaTeX source, generate a PDF from it, upload the PDF to Commons, and enter the LaTeX sources into the text part of the file on Commons. And then, anyone who may want to modify the PDF will have access to the source and there will be revision history for the source. I am not using LaTeX (and Beamer?), but I used an analogous process for Python code and SVG: Python code is the source and the SVG is the presentation-layer artifact generated from the source. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:47, 3 November 2025 (UTC)
::::Great idea! I’ll try that :-), thanks. I wanted to focus on adding materials that are easier to modify in the future and also are lighter (less data heavy). Many thanks Dan. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:47, 3 November 2025 (UTC)

== Help us decide the name of the new Abstract Wikipedia project ==

&lt;section begin=&quot;function1&quot;/&gt;
{{int:Hello}}. Please help pick a name for the new Abstract Wikipedia wiki project. This project will be a wiki that will enable users to combine functions from [[:f:|Wikifunctions]] and data from Wikidata in order to generate natural language sentences in any supported languages. These sentences can then be used by any Wikipedia (or elsewhere).

There will be two rounds of voting, each followed by legal review of candidates, with votes beginning on 20 October and 17 November 2025. Our goal is to have a final project name selected on mid-December 2025. If you would like to participate, then '''[[m:Special:MyLanguage/Abstract Wikipedia/Abstract Wikipedia naming contest|please learn more and vote now]]''' at meta-wiki.
{{Int:Feedback-thanks-title}}
&lt;section end=&quot;function1&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 11:43, 20 October 2025 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on October 30, 2025. Applications for the Affiliations Committee, Ombuds commission and the Case Review Committee close on December 11, 2025. Learn how to apply by [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Committee appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email cst[[File:At sign.svg|16x16px|link=|(_AT_)]]wikimedia.org with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-[[m:User:MKaur (WMF)| MKaur (WMF)]] 14:13, 30 October 2025 (UTC)
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      <sha1>9g7ft6k87jpatoxtaqh36h6oy2szx9s</sha1>
    </revision>
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      <comment>/* Wikiversity:Curators to become a policy */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:11, 3 April 2026 (UTC)
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:I didnt find such properties, so we are probably fine. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:00, 12 April 2026 (UTC)
:: +1 (agreed). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:19, 12 April 2026 (UTC)

== Advice needed: A Neurodiversity-inspired Idea/observation ==

If I want the greatest participation of others to &quot;provide constructive criticism to my idea&quot; or to &quot;shoot down my idea&quot; or &quot;idea&quot;.

What I've called it so far is &quot;The Neurodiversity-inspired Idea&quot;. At other times I used more sensationalist wording but here on Wikiversity I don't dare do that. I actually woke up with thinking about putting this into my userspace draft: &quot;Personal Observations Made By Meeting Autistic and Non-Autistic Adults&quot;.

My ultimate goal is to stop blathering about my &quot;idea&quot; to friend and family without feeling my &quot;methodology&quot; is going into any progressive direction whatsoever. My latest encounter was somewhat constructive though. A friend of a friend who worked with people presenting ideas in attempting to getting grants. I don't want a grant. I just want to figure out how I can express my &quot;idea&quot; in a way so that I can more clearly figure out what flaws it got.

At the same time I tend to overthink. If anyone thinks etherpad might be a good place and considering Wikimedia already got an etherpad at https://etherpad.wikimedia.org/ if anyone feels like they know me better in the future feel free to suggest a &quot;session&quot; on etherpad.

'''If I don't receive a reply to this in 1 week's time I will begin to explore this &quot;idea&quot; into my userspace''' unless you replied and refrained me from doing so, of course. Then maybe after &quot;developing it there&quot; I might reference it to you another future time here in the Colloquium, with my &quot;idea&quot; still in my userspace draft. This &quot;idea&quot; is sort of a burden, I'm happy I've made the choice to get rid of it and hopefully move on with my life, unless there is something to this &quot;idea&quot;. 

My failure is probably evident: I feel I haven't told you anything. Same happened to when I talked to friends and family. In danger of overthinking it further I'll publish this right now. I need to &quot;keep it together&quot; [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:36, 16 April 2026 (UTC)

:Good on you putting it out there ... and hitting publish :). I'd say go for it (no need to wait), give birth to your idea and share about it here and elsewhere. Let it take shape and see where it might go. In many ways, this is exactly what an open collaborative learning community should be doing. Others might not know well how to respond, so perhaps consider creating some questions to accompany the idea. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 16 April 2026 (UTC)
::Thank you for encouraging me in developing the idea.
::I have created a &quot;questions&quot; section in the draft which is visible in the table of contents now. My brain was &quot;frozen&quot; today metaphorically speaking in that I felt I had like a &quot;writer's block&quot; so the draft has more &quot;AI/LLM&quot; content than before. I used the LLM for generating questions. The answers are so far human-only.
::I've also created a subsection where I could add the prompts that made the LLM generate the questions. That could help people make better prompts perhaps. I've described what it is about inside of it and there are some chaotically written notes.
::[[Draft:The_Neurodiversity-inspired_Idea#Questions_that_might_encourage_the_development_of_this_idea_and_its_methodology]]
::My draft is missing stuff. Any questions that you contribute to my draft will probably help me and if I don't understand the questions I'll probably notify you and also at the same time &quot;feed them&quot; to an LLM and ask in my input like &quot;explain in simple words what this question means, what is it searching for?&quot; etc. while I wait for an answer. If you have any more feedback please give it to me here or on the Draft page, its talk page or my user talk page. Thank you for helping me! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:20, 18 April 2026 (UTC)
::Today I woke up with not only thinking about supplying questions along with the &quot;idea&quot; but also answers. ie. Is it possible to &quot;test&quot; this idea? Is it possible to create one or multiple hypotheses based on this &quot;idea&quot;?(etc.) I've thought about this before in this &quot;idea&quot; but since I'm beginning to add to Wikiversity what was previously 'locked in my mind' it's also easier for me to see what I've done so far. Thank you for this comment! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 09:11, 23 April 2026 (UTC)
:May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish. Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet. Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible. Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents. --  [[User:Eric.LEWIN|Eric.LEWIN]] ([[User talk:Eric.LEWIN|discussion]] • [[Special:Contributions/Eric.LEWIN|contributions]]) 10:06, 17 April 2026 (UTC)
::Sorry about the false positive on the profanity filter - I've fixed it. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:26, 17 April 2026 (UTC)
:::&quot;May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish.&quot;
::Thank you Eric for this comment. Trust in time is how I interpret it. I should not feel like I need to be in a hurry. I'll try to give this time. Thank you!
:::&quot;Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet.&quot;
::A central place for developing or making &quot;project notes&quot; regarding the Neurodiversity idea on my userspace, I might need that, like a diary or &quot;project notes&quot; of the Neurodiversity idea similar to my course notes regarding my experience with Coursera.
::Any actions I take are going to be related to my Userspace from now on but I'll also update the draft when necessary. Now in the beginning I might be working daily to once every 3 days on both the draft and the daily notes I plan to make.
:::&quot;Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible.&quot;
::Thank you for the advice. I was brainstorming yesterday about it. I concluded that since I've not yet developed a methodology that adheres to &quot;Do no harm&quot; and this is my first time working my &quot;idea&quot; into a way that is compatible with how projects develop on English Wikiversity this is new to me. My methodology isn't developed and therefore trying to get attention to my project through a name can wait. Yesterday I figured out a silly title that has nothing to do with the project: &quot;Planetary Awareness Potato Cabbage Rolls&quot; or something like that. Google output read that no such thing exists so I wanted it mainly to be unique. I don't want to raise attention that I'm unsure whether I'll actually be capable of developing a methodology for but project notes is my best bet so far in tracking my progress. Every day I think about this &quot;idea&quot; but I need to improve the important parts.
:::&quot;Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents.&quot;
::You added great points and I felt that I was helped by you! I encourage you to post again and I can understand that interacting with any kind of automated filter can be discouraging and can be for me too! Thank you for giving me feedback! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 16:01, 18 April 2026 (UTC)

== Enable the abuse filter block action? ==

In light of [[Special:AbuseLog/80178]] (coupon spam), I would like to propose enabling the block action for the abuse filter. Only custodians will be able to enable and disable that action on an abuse filter, and it is useful to block ongoing vandalism. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:12, 13 April 2026 (UTC)

:Seems like a good idea, almost all of the users which create such pages are spambots so this shouldn’t be a problem. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:41, 13 April 2026 (UTC)
:Can you explain some more (I am new to abuse filters)? It looks like the attempted edit was prevented? Which abuse filter?
:Note on your suggestion, have also reactivated Antispam Filter 12 - see [[WV:RCA]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:45, 15 April 2026 (UTC)
:: I am proposing that we activate the abuse filter block action, which if a user triggers an abuse filter, it would actually block the user in question - the same mechanism that a custodian would use to block users. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:11, 15 April 2026 (UTC)
:::OK, thankyou, that makes sense. And, reviewing the abuse filter 12 log, it would be helpful because it would prevent the need for manual blocking. But I don't see a setting for autoblocking? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:14, 15 April 2026 (UTC)
:::: I think it probably adds an autoblock. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:43, 16 April 2026 (UTC)
: [[User:Jtneill|Jtneill]] and [[User:PieWriter|PieWriter]], given that a little bit more than a week has passed and there is minimal consensus to activate the abuse filter block action, I filed [[phab:T424053]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:05, 21 April 2026 (UTC)
::Thank-you for doing this. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:03, 24 April 2026 (UTC)

== Wikiversity:Curators to become a policy ==
{{archive top|There is strong consensus, so [[Wikiversity:Curators]] is now a policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:16, 9 May 2026 (UTC)}}
I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)
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      <comment>/* Interface administrator for Codename Noreste */ archive from [[Wikiversity:Colloquium]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
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== Interface administrator for Codename Noreste ==
{{Archive top|After running for a week, there is clear consensus for [[User:Codename Noreste]] to have Interface admin rights for 120 days; implemented until 10 September, 2026 -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 12 May 2026 (UTC)}}

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)

:{{Comment}} Could @[[User:Codename Noreste|Codename Noreste]] delete [[MediaWiki:Gadget-WikiSign.js]], which was requested to be deleted @[[User:Koavf|Justin]], @[[User:Jtneill|Jtneill]], @[[User:Atcovi|Atcovi]]? I dont think we need it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:40, 9 May 2026 (UTC)
::Yes - clearly no longer used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:18, 9 May 2026 (UTC)
::: I can't delete it because I don't have the required permissions to do so.
::: On a side note, if this project has a need for permanent interface administrators, I would suggest that we have a minimum of two IAs, similar to how there must be two CUs and/or suppressors (or none). Maybe Koavf can be a good candidate if I am elected for permanent interface adminship, and I believe that permission shouldn't be removed from someone's own account. Instead, a bureaucrat should do it. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:20, 9 May 2026 (UTC)
::::I am willing and happy to do it, unfortunately, we do not have an appetite for indef IAs and just had a discussion that resulted in a [https://en.wikiversity.org/w/index.php?title=Wikiversity:Interface_administrators&amp;diff=prev&amp;oldid=2807543 consensus that we can have IAs that have the user rights for 14 to 120 days]. So once you have the rights, please make sure to gopher it. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:54, 9 May 2026 (UTC)
:::::@[[User:Koavf|Koavf]] give it time. Look at me, I was in favor of shorter time, now I am looking back to times, when custodians could do it without the need of extra flag. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:31, 9 May 2026 (UTC)
::::::Here's hoping. I think it would reduce administrative overhead, but that's just me and I'm not a bureaucrat here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:33, 9 May 2026 (UTC)
::::Complicated. Where are the times, admins could do everything! [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:27, 9 May 2026 (UTC)
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== [[Wikiversity:Bureaucratship]] to become a policy ==
{{archive top|'''Approved - now a policy'''. 5 supports + 1 nominator. No objections.}}
Following the recent approval of [[Wikiversity:Curators]] as a policy, I think [[Wikiversity:Bureaucratship]] may also be ready for policy status.

Please share your views about whether bureaucratship is ready to become a policy, or whether further revisions are needed.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:58, 9 May 2026 (UTC)
: I added a logo about that user group, but other than that, it looks good to me. {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:38, 9 May 2026 (UTC)
:I think that the consensus on this policy is proven by years of using it without further changes. But I I have to say weather I agree with this to become a policy, than of course {{support}}. It works and there were no major issues with it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:45, 9 May 2026 (UTC)
:{{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:51, 10 May 2026 (UTC)
:{{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:37, 11 May 2026 (UTC)
:{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:51, 12 May 2026 (UTC)
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    <title>Communications Law in Spain</title>
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== '''&lt;big&gt;Communication Law in Spain&lt;/big&gt;''' ==

===== Introduction =====
Spain is a parliamentary constitutional monarchy.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1978-31229#:~:text=24%3A%20%23a18%5D-,Art%C3%ADculo%2018,2.|title=BOE-A-1978-31229 Constitución Española.|website=www.constituteproject.org|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:36&quot;&gt;{{Cite web|url=https://european-union.europa.eu/principles-countries-history/eu-countries/spain_en|title=Spain – EU country {{!}} European Union|website=european-union.europa.eu|language=en|access-date=2026-03-05}}&lt;/ref&gt; While the monarch serves as head of state, political power is exercised through a democratic parliamentary system led by a prime minister and the national legislature known as the Cortes Generales.&lt;ref&gt;{{Cite web|url=https://www.lamoncloa.gob.es/lang/en/espana/leyfundamental/paginas/titulo_tercero.aspx|title=Part III The Cortes Generales|website=www.lamoncloa.gob.es|language=en|access-date=2026-05-03}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.senado.es/web/conocersenado/temasclave/cortesgenerales/index.html?lang=en|title=Parliament (Cortes Generales) - Spanish Senate (official website)}}&lt;/ref&gt; Spain is also a highly decentralized state composed of seventeen Autonomous Communities, each with its own regional government and authority over areas such as culture, language policy, and public broadcasting.&lt;ref name=&quot;:0&quot; /&gt; In addition to its domestic institutions, Spain operates within a broader European legal framework as a member of the European Union and a party to international human rights agreements such as the European Convention on Human Rights.&lt;ref name=&quot;:36&quot; /&gt;

Modern debates over communication law in Spain are also deeply influenced by the country’s twentieth-century history. From 1939 until 1975 Spain was governed by the authoritarian dictatorship of General Francisco Franco, during which the state exercised strict control over political speech and media institutions.&lt;ref&gt;{{Cite web|url=https://www.history.com/articles/francisco-franco|title=Francisco Franco - Biography, Facts &amp; Death|last=Editors|first=HISTORY com|date=2009-11-09|website=HISTORY|language=en|access-date=2026-03-05}}&lt;/ref&gt; Following Franco’s death, Spain underwent a democratic transition that culminated in the adoption of the 1978 Constitution, which established modern protections for freedom of expression and democratic pluralism.&lt;ref&gt;{{Cite web|url=https://adst.org/2016/06/spains-post-franco-emergence-dictatorship-democracy/|title=Spain’s Post-Franco Emergence from Dictatorship to Democracy – Association for Diplomatic Studies &amp; Training|language=en-US|access-date=2026-03-05}}&lt;/ref&gt; These historical experiences continue to shape contemporary debates over speech, protest, and public memory in Spain.

== &lt;big&gt;Sources and Institutions Of Communication Law In Spain&lt;/big&gt; ==

=== '''National Sources and Institutions''' ===

===== Constitutional Foundations of Communication Law =====
The Spanish Constitution of 1978 is the supreme legal authority governing communication rights in Spain.&lt;ref name=&quot;:0&quot; /&gt; Only a few provisions directly address communication, but they shape disputes involving the press, privacy, defamation, surveillance, and protest.&lt;ref name=&quot;:0&quot; /&gt;

The first major provision is Article 20, which protects freedom of expression and information.&lt;ref name=&quot;:0&quot; /&gt; It guarantees freedom of expression, creative and academic freedom, the right to communicate and receive truthful information, and the prohibition of prior censorship.&lt;ref name=&quot;:0&quot; /&gt; This is the backbone of Spanish communication law.

But Article 20 is not a blank check. Article 20(4) makes clear that expression has limits when it collides with other constitutional rights. &lt;ref name=&quot;:0&quot; /&gt; In other words, Spain builds speech protection and speech limits into the same constitutional design.

That leads to the second key provision: Article 18, which protects privacy, honor, and the secrecy of communications. Article 18 expressly protects the right to honor, personal and family privacy, personal image, and the secrecy of communications.&lt;ref name=&quot;:0&quot; /&gt; These protections frequently arise in modern communication disputes. For example, in Spanish Constitutional Court decision STC 104/1986, the court examined whether a newspaper report accusing a businessman of misconduct violated his constitutional right to honor, emphasizing the need to balance expression with protection of reputation.&lt;ref&gt;{{Cite web|url=https://hj.tribunalconstitucional.es/en/Resolucion/Show/104|title=HJ System - Decision: SENTENCIA 62/1982|website=hj.tribunalconstitucional.es|access-date=2026-03-05}}&lt;/ref&gt;

Another important principle in the Spanish constitutional system is the protection of the “essential content” of fundamental rights, often referred to as the ''núcleo esencial''. Rooted in Article 10 and Section I on fundamental rights, this principle holds that certain core aspects of rights cannot be undermined by the state.&lt;ref&gt;{{Cite web|url=https://www.lamoncloa.gob.es/lang/en/espana/leyfundamental/paginas/titulo_primero.aspx|title=Part I Fundamental Rights and Duties|website=www.lamoncloa.gob.es|language=en|access-date=2026-03-05}}&lt;/ref&gt; The doctrine reflects Spain’s constitutional commitment to human dignity and the free development of personality. In practice, rights may be regulated but not restricted in ways that destroy their core substance. Rights such as expression, life, and physical integrity retain a protected core beyond ordinary political decision-making.

An interesting wrinkle in the Spanish Constitution is Article 10(2), often called the international interpretation clause.&lt;ref name=&quot;:0&quot; /&gt; It requires that constitutional rights be interpreted in conformity with international human rights treaties ratified by Spain. That strengthens the influence of European and international human-rights standards inside Spain’s own constitutional system. For example, in ''Stern Taulats and Roura Capellera v. Spain'' (2018), the European Court of Human Rights ruled that Spain violated freedom of expression after protesters were convicted for burning photographs of the King during a political demonstration, illustrating how international courts shape constitutional speech protections.&lt;ref&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22001-181719%22%5D%7D|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-05}}&lt;/ref&gt;

===== Regulatory Authorities =====
Spain relies on regulatory authorities to implement and supervise communication law.

The National Commission on Markets and Competition (CNMC) oversees telecommunications and audiovisual markets in Spain, with a role that blends sector oversight with competition regulation.&lt;ref&gt;{{Cite web|url=https://www.cnmc.es/|title=Comisión Nacional de los Mercados y la Competencia {{!}} CNMC|website=www.cnmc.es|access-date=2026-03-02}}&lt;/ref&gt;

The Spanish Data Protection Agency (AEPD) enforces the GDPR and Organic Law 3/2018, and it is one of the main places where “digital rights” become real—through guidance, enforcement, and sanctions.&lt;ref&gt;{{Cite web|url=https://www.aepd.es/|title=Agencia Española de Protección de Datos {{!}} AEPD|website=www.aepd.es|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2018-16673|title=BOE-A-2018-16673 Ley Orgánica 3/2018, de 5 de diciembre, de Protección de Datos Personales y garantía de los derechos digitales.|website=www.boe.es|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://digital.gob.es/|title=Regulation - 2016/679 - EN - gdpr - EUR-Lex|website=eur-lex.europa.eu|language=en|access-date=2026-03-02}}&lt;/ref&gt;

The Ministry for Digital Transformation and Public Function plays a coordinating role for national telecommunications and digital policy, including the domestic implementation of EU digital regulation. The ministry also oversees the allocation of radio frequencies, a critical responsibility because the radio spectrum is a limited public resource used by mobile networks, television broadcasting, satellite communications, and other wireless technologies.&lt;ref name=&quot;:12&quot;&gt;{{Cite web|url=https://digital.gob.es/|title=Portal MTDFP {{!}} Inicio|website=digital.gob.es|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:9&quot; /&gt;

===== National Legislative Framework =====
Spain does not rely solely on the Constitution and international treaties to regulate communication.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://www.echr.coe.int/european-convention-on-human-rights|title=European Convention on Human Rights (ECHR) - EUR-Lex|date=2009-12-01|website=eur-lex.europa.eu|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:11&quot; /&gt; Spain can pass national legislation governing communication as long as it stays consistent with superior constitutional and supranational law.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://curia.europa.eu/site/jcms/d2_5093/en/the-court-of-justice|title=Court of Justice of the European Union|website=curia|language=en|access-date=2026-03-02}}&lt;/ref&gt;

The General Audiovisual Communication Law (Law 13/2022) regulates television, radio, and on-demand audiovisual services, including licensing, protection of minors, advertising standards, and media pluralism.&lt;ref name=&quot;:1&quot;&gt;{{Citation|title=Ley 13/2022, de 7 de julio, General de Comunicación Audiovisual|url=https://www.boe.es/eli/es/l/2022/07/07/13|date=2022-07-08|accessdate=2026-03-02|pages=96114–96220|issue=Ley 13/2022|last=Jefatura del Estado}}&lt;/ref&gt; It also functions as Spain’s main implementation of AVMSD requirements.&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;

The General Telecommunications Law (Law 11/2022) regulates electronic communications networks and services, including spectrum allocation and operator licensing.&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://ppp.worldbank.org/library/general-de-telecomunicaciones-ley-11-2022|title=General de Telecomunicaciones Ley 11/2022|website=PUBLIC-PRIVATE-PARTNERSHIP LEGAL RESOURCE CENTER|language=en|access-date=2026-03-02}}&lt;/ref&gt; Under Article 149.1.21 of the Constitution, telecommunications is an exclusive competence of the State.&lt;ref name=&quot;:0&quot; /&gt; In other words, national control ensures consistent regulation of telecommunications across Spain’s 17 Autonomous Communities.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot; /&gt;

On privacy, Spain applies the EU’s General Data Protection Regulation (GDPR) and complements it through Organic Law 3/2018 (LOPDGDD), which regulates data processing and sets out digital rights in domestic law.&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt; This framework includes digital rights such as the “right to erasure” (“right to be forgotten”).&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:12&quot; /&gt;

Finally, Organic Law 1/1982 on the Protection of Honor, Privacy, and Personal Image provides civil remedies when freedom of expression conflicts with personal dignity, basically, when speech unlawfully harms reputation or private life.&lt;ref name=&quot;:13&quot;&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1982-11196|title=BOE-A-1982-11196 Ley Orgánica 1/1982, de 5 de mayo, de protección civil del derecho al honor, a la intimidad personal y familiar y a la propia imagen.|website=www.boe.es|access-date=2026-03-02}}&lt;/ref&gt; This law operationalizes the protections in Article 18 in everyday disputes involving media reporting and personal reputation.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

===== Regional (Autonomous Community) Regulation =====
[[File:Autonomous communities of Spain no names-gl.svg|thumb|'''The 17 Autonomous communities of Spain''']]
Spain is a decentralized state composed of 17 Autonomous Communities. While telecommunications remains a national competence under Article 149, Autonomous Communities still influence communication in meaningful ways, especially through public broadcasting and language policy.&lt;ref name=&quot;:0&quot; /&gt;

Autonomous Communities may create and regulate their own public broadcasting corporations. For example, Catalonia operates the Corporació Catalana de Mitjans Audiovisuals (CCMA)&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://www.3cat.cat/corporatiu/en/el-grup/|title=The Group - CCMA|last=3Cat|website=3Cat|language=en|access-date=2026-03-02}}&lt;/ref&gt; and the Basque Country operates EITB.&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://www.eitb.eus/en/|title=EITB|website=www.eitb.eus|language=en|access-date=2026-03-02}}&lt;/ref&gt; These bodies operate under regional frameworks but still sit under national and EU rules that shape audiovisual services more generally.&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:10&quot; /&gt;

Some regions also maintain audiovisual supervisory authorities. Catalonia, for example, has the Consell de l’Audiovisual de Catalunya (CAC), which oversees audiovisual services within the region and has a particular focus on standards tied to language and culture.&lt;ref&gt;{{Cite web|url=https://www.cac.cat/|title=Consell de l'Auidovisual de Catalunya}}&lt;/ref&gt;

Regional governments also regulate language and cultural policy. Autonomous Communities with co-official languages may adopt measures that promote regional-language media and broadcasting quotas.&lt;ref name=&quot;:0&quot; /&gt; These policies shape what audiences actually see and hear day-to-day, but they still must remain consistent with Spain’s constitutional protections and EU standards.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:8&quot; /&gt;

=== '''International''' '''Sources and Institutions''' ===

===== European Union Law =====
	As an EU Member State, Spain is bound by European Union law, including the principle that EU law has primacy in areas where the EU has competence.&lt;ref name=&quot;:4&quot; /&gt; EU rules increasingly shape digital communication and audiovisual markets.&lt;ref name=&quot;:8&quot;&gt;{{Citation|title=Charter of Fundamental Rights of the European Union|url=http://data.europa.eu/eli/treaty/char_2012/oj/eng|date=2012-10-26|accessdate=2026-03-02|language=en}}&lt;/ref&gt;&lt;ref name=&quot;:9&quot;&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2022/2065/oj/eng|title=Regulation - 2022/2065 - EN - DSA - EUR-Lex|website=eur-lex.europa.eu|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:10&quot;&gt;{{Citation|title=Directive 2010/13/EU of the European Parliament and of the Council of 10 March 2010 on the coordination of certain provisions laid down by law, regulation or administrative action in Member States concerning the provision of audiovisual media services (Audiovisual Media Services Directive) (Codified version) (Text with EEA relevance)|url=http://data.europa.eu/eli/dir/2010/13/oj/eng|date=2010-03-10|accessdate=2026-03-02|volume=095|language=en}}&lt;/ref&gt;

Article 11 of the EU Charter of Fundamental Rights protects freedom of expression and media pluralism.&lt;ref name=&quot;:8&quot; /&gt; When acting within EU law, Spanish authorities must comply with these protections.

Two major EU instruments show how direct this influence can be. First, the Digital Services Act (Regulation (EU) 2022/2065) regulates online platforms and intermediary services across the EU, with transparency duties, processes for handling illegal content, and heightened obligations for very large online platforms.&lt;ref name=&quot;:9&quot; /&gt; Spain must enforce these rules through its national system.

Second, the Audiovisual Media Services Directive (AVMSD) sets EU-wide standards for television and on-demand audiovisual services, including advertising rules, protections for minors, and promotion of European content.&lt;ref name=&quot;:10&quot; /&gt; Spain’s General Audiovisual Communication Law (2022) implements these European requirements in national law.&lt;ref name=&quot;:1&quot; /&gt;

===== International Obligations =====
	Spain is also a party to major international human rights treaties that shape communication law.

Spain participates in the World Intellectual Property Organization (WIPO), a specialized agency of the United Nations that administers international systems for protecting intellectual property. WIPO maintains global databases for searching patents, trademarks, and industrial designs across jurisdictions.&lt;ref&gt;{{Cite web|url=https://www.wipo.int/|title=WIPO - World Intellectual Property Organization|website=www.wipo.int|language=en|access-date=2026-03-05}}&lt;/ref&gt; For example, the PATENTSCOPE database allows users to search millions of international patent applications filed under the Patent Cooperation Treaty, while the Global Brand Database provides access to trademark records from national and international registries.&lt;ref&gt;{{Cite web|url=https://patentscope.wipo.int/search/en/search.jsf|title=WIPO - Search International and National Patent Collections|website=patentscope.wipo.int|access-date=2026-03-05}}&lt;/ref&gt; These tools help prevent conflicting claims and support cross-border protection of intellectual property.

Spain is also a party to the International Covenant on Civil and Political Rights (ICCPR), which protects freedom of expression in Article 19.&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://www.ohchr.org/en/instruments-mechanisms/instruments/international-covenant-civil-and-political-rights|title=International Covenant on Civil and Political Rights|website=OHCHR|language=en|access-date=2026-03-02}}&lt;/ref&gt; Because of Article 10(2) of the Spanish Constitution, Spain’s courts must read domestic constitutional rights consistently with these kinds of international commitments.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt;

A key upshot of this layered legal system is that freedom of expression in Spain is not at the mercy of the national political process alone.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; Because Spain operates within a broader European and international legal order, attempts to narrow expression face external legal constraints.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:11&quot; /&gt; This layered system makes it less likely that core expressive freedoms will be reduced.

== &lt;big&gt;Freedom of Expression and Dignity in Spain&lt;/big&gt; ==

===== Constitutional Balance: Expression and Honor =====
Spain protects freedom of expression under Article 20 of the 1978 Constitution, which guarantees the right to express and disseminate ideas and to communicate and receive truthful information, while prohibiting prior censorship.&lt;ref name=&quot;:0&quot; /&gt; At the same time, Article 18 protects the right to honor, privacy, and personal image, protections that are further implemented through Organic Law 1/1982 on the Protection of Honor, Privacy, and Personal Image.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; The substance of these two provisions often collide, especially because Spanish courts treat them as equally serious constitutional commitments.

Unlike systems that treat speech as nearly absolute, Spain’s Constitutional Court uses a balancing approach. When expression conflicts with dignity or reputation, courts weigh the competing rights while ensuring that the essential content (''núcleo esencial'') of each constitutional right is preserved, meaning that neither freedom of expression nor the protection of honor and privacy may be restricted in a way that destroys their core substance.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt; In practice, this can mean allowing strong criticism of public officials or institutions when it contributes to democratic debate, while still permitting legal remedies when speech crosses into false factual allegations or serious attacks on personal reputation. Spanish constitutional jurisprudence has repeatedly emphasized that freedom of expression has a “preferred position” in democratic debate, especially when speech concerns political issues or public officials.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:5&quot; /&gt; But that preferred position does not make it untouchable.

This framework reflects Spain’s transition to democracy after the Franco dictatorship.&lt;ref&gt;{{Cite web|url=https://www.realinstitutoelcano.org/en/work-document/international-dimensions-of-democratisation-revisiting-the-spanish-case/|title=International dimensions of democratisation: revisiting the Spanish case|last=Powell|first=Charles|website=Elcano Royal Institute|language=en-US|access-date=2026-03-02}}&lt;/ref&gt; The 1978 Constitution placed strong emphasis on open public debate as essential to pluralism.&lt;ref name=&quot;:0&quot; /&gt; At the same time, dignity is considered a foundational value of the constitutional order. This dual commitment to democratic openness and protection of personal honor defines Spain’s speech doctrine.

===== The Importance of Veracity =====
One distinctive feature of Spanish law is the requirement of “veracity.” Veracity in ethics is the principle of truth-telling, requiring professionals to be honest, transparent, and accurate in all communications to foster trust. Article 20 protects the right to communicate “truthful information.”&lt;ref name=&quot;:0&quot; /&gt; Courts do not interpret this to mean that journalists must prove absolute truth.&lt;ref name=&quot;:13&quot; /&gt; Instead, they must show that they acted with reasonable diligence in verifying their information.

This standard recognizes human limits: reporters and witnesses cannot know “the whole truth.” What matters is whether they checked reliable sources and acted in good faith. If they do, even mistaken reporting may still be protected. If they fail to verify serious factual claims that harm someone’s reputation, liability may follow.&lt;ref name=&quot;:13&quot; /&gt;

The Constitutional Court has distinguished sharply between opinions and factual statements.&lt;ref name=&quot;:13&quot; /&gt; Opinions, especially political opinions, receive strong protection, even when harsh or offensive. Factual allegations that damage someone’s honor are treated differently. In defamation cases, courts examine whether the information contributed to public debate or merely harmed reputation without public interest.&lt;ref name=&quot;:13&quot; /&gt;

===== Terrorism, the Monarchy, and Controversial Speech =====
The limits of Spain’s balancing approach become most visible in politically sensitive cases.

Following decades of violence by the Basque terrorist group ETA, Spain criminalized the glorification of terrorism and humiliation of victims under Article 578 of the Criminal Code.&lt;ref&gt;{{Cite web|url=https://www.wipo.int/wipolex/en/legislation/details/18760|title=Penal Code (Organic Law No. 10/1995 of November 23, 1995, as amended up to Organic Law No. 2/2019 of March 1, 2019), Spain, WIPO Lex|website=www.wipo.int|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:14&quot;&gt;{{Cite web|url=https://fibgar.es/en/the-human-rights-committee-urges-spain-to-protect-freedom-of-expression-and-human-rights-defenders/|title=The Human Rights Committee urges Spain to protect freedom of expression and human rights defenders|last=Fibgar|date=2025-08-06|website=FIBGAR|language=en-US|access-date=2026-03-02}}&lt;/ref&gt; Supporters argue that these laws protect democratic stability and the dignity of victims. Critics argue that they have sometimes been applied too broadly, including against musicians and social media users.&lt;ref name=&quot;:15&quot;&gt;{{Cite web|url=https://www.amnesty.org/en/documents/eur41/001/2014/en/|title=Spain: The right to protest under threat|date=2014-04-24|website=Amnesty International|language=en|access-date=2026-03-02}}&lt;/ref&gt;

In Otegi Mondragón v. Spain (2011), a Basque politician was convicted for referring to the King as the “chief of the torturers.”&lt;ref name=&quot;:16&quot;&gt;{{Cite web|url=https://hudoc.echr.coe.int/spa?i=001-103951|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-02}}&lt;/ref&gt; Spain’s courts upheld the conviction, but the European Court of Human Rights ruled that the conviction violated freedom of expression under Article 10 of the European Convention on Human Rights. The Strasbourg court emphasized that political speech, even when provocative, deserves heightened protection and that public institutions must tolerate stronger criticism.&lt;ref name=&quot;:16&quot; /&gt;

A similar controversy arose in Stern Taulats and Roura Capellera v. Spain (2018), involving protesters who burned photographs of the King during a political demonstration.&lt;ref name=&quot;:17&quot;&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng?i=001-181724|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-02}}&lt;/ref&gt; Spanish courts treated the act as an insult to the Crown. The European Court again ruled that Spain had violated freedom of expression, finding that the act was symbolic political protest rather than incitement to violence.&lt;ref name=&quot;:17&quot; /&gt;

Artistic expression has also generated debate. The prosecution of rappers such as Valtonyc for lyrics praising terrorist groups or insulting state institutions sparked international criticism.&lt;ref&gt;{{Cite web|url=https://arisa-project.eu/the-presumption-of-innocence-and-the-media-coverage-of-criminal-cases/|title=The Presumption of Innocence and the Media Coverage of Criminal Cases|last=admin|date=2021-05-13|website=Arisa|language=en-US|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:18&quot;&gt;{{Cite web|url=https://njc.dk/wp-content/uploads/2018/04/Putting-the-chill-in-media-freedom-and-free-speech-.pdf|title=Putting	the	chill	in media	freedom	and	free	speech}}&lt;/ref&gt;Some observers argued that criminal sanctions risked chilling artistic freedom.&lt;ref name=&quot;:18&quot; /&gt; Others defended the prosecutions as necessary to prevent normalization of violence.&lt;ref name=&quot;:14&quot; /&gt;

These cases reveal a deeper tension in Spain over how far a democracy can go in protecting institutional dignity and social peace without narrowing the space for dissent.

===== Ongoing Debate: Dignity-Centered Democracy =====
Spain’s speech model is often described as dignity-centered. Human dignity is explicitly recognized in Article 10 of the Constitution as a foundational principle of the legal order.&lt;ref name=&quot;:0&quot; /&gt; Courts therefore treat attacks on honor, reputation, or institutional integrity as constitutionally significant.&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:13&quot; /&gt;

Some scholars argue that this model reflects a mature constitutional democracy that refuses to sacrifice personal dignity in the name of absolute speech.[35][36] They see Spain’s approach as consistent with broader European human rights traditions, where proportionality and balancing are central.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;

Others argue that criminal penalties for offensive speech, especially in political or artistic contexts, create a chilling effect and discourage open debate.&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt; They point to repeated rulings from the European Court of Human Rights pushing Spain toward stronger protection of political expression.&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt;

Spain’s doctrine continues to evolve through judicial dialogue between national courts and European institutions.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;The result is a system that seeks to protect democratic debate while also preserving the constitutional value of dignity, a balance that remains contested and actively debated.

== &lt;big&gt;Spain’s 2015 Citizen Security Law (“Gag Law”)&lt;/big&gt; ==
[[File:Manifestación contra la Ley Mordaza en Madrid 20-12-2014 - 07.jpg|thumb|On December 20, 2014, protesters in Madrid demonstrated against Spain’s new Citizens Security Law, known as the &quot;Gag Law&quot; (Ley Mordaza)]]
The Citizen Security Law (Ley Orgánica 4/2015 de protección de la seguridad ciudadana) is a Spanish national law that entered into force on 1 July 2015.&lt;ref name=&quot;:19&quot;&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2015-3442|title=BOE-A-2015-3442 Ley Orgánica 4/2015, de 30 de marzo, de protección de la seguridad ciudadana.|website=www.boe.es|access-date=2026-03-02}}&lt;/ref&gt; It is widely known in public debate as the “Gag Law” (Ley Mordaza), a nickname given by critics who argue that it discourages protest and limits free expression through financial penalties rather than formal censorship.&lt;ref name=&quot;:20&quot;&gt;{{Cite web|url=https://www.hrw.org/news/2015/03/09/spain-reject-flawed-public-security-bill|title=Spain: Reject Flawed Public Security Bill {{!}} Human Rights Watch|date=2015-03-09|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:21&quot;&gt;{{Cite web|url=https://www.amnesty.org/en/latest/news/2015/03/spain-two-pronged-assault-targets-rights-and-freedoms/|title=Spain: Two-pronged assault targets rights and freedoms of Spanish citizens, migrants and refugees|date=2015-03-26|website=Amnesty International|language=en|access-date=2026-03-02}}&lt;/ref&gt;

The law was introduced by Spain’s government as a modernization of public-order regulations. Officials stated that it was designed to provide clearer rules for police operations, maintain public security, and respond to disruptive protest activity. Supporters emphasize that the law does not establish prior censorship and does not criminalize political opinions as such.&lt;ref name=&quot;:19&quot; /&gt;

Critics, however, argue that while the Constitution only prohibits prior censorship, the Gag Law creates a system of administrative fines imposed after expression, which can still discourage participation in protests and public criticism. They contend that heavy fines can have a chilling effect, especially on journalists and activists.&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:21&quot; /&gt;

===== Key Provisions of The Citizen Security Law =====
The Citizen Security Law establishes a detailed system of administrative infractions and sanctions. Fines range from several hundred euros to up to €600,000 in the most serious cases.&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:34&quot;&gt;{{Cite news|url=https://www.theguardian.com/world/2015/mar/12/spain-security-law-protesters-freedom-expression|title=Spain puts 'gag' on freedom of expression as senate approves security law|last=Kassam|first=Ashifa|date=2015-03-12|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}&lt;/ref&gt;

Among the most controversial provisions are:

* Fines (up to €600) for holding public demonstrations without prior notification, even if peaceful
* Fines (up to €600) for protests that deviate from approved routes
* Fines (up to €30,000) for protests causing disturbances near Parliament or regional government buildings
* Fines (up to €600,000) for unauthorized protests near key infrastructure (airports, nuclear plants, refineries, transport hubs)
* Fines (up to €30,000) for obstructing police or officials carrying out evictions or court orders
* Fines (up to €30,000) for recording or publishing images of police officers if deemed to endanger their safety or an operation&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:34&quot; /&gt;

Human rights organizations have argued that the wording of some provisions is broad and gives authorities significant discretion in enforcement.&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:21&quot; /&gt;

===== International Reaction =====
The “Gag law” was met with strong criticism from international human rights groups even before it entered into force. Human Rights Watch warned that the legislation undermined freedom of assembly and expression by allowing heavy fines for peaceful protest and for recording police conduct.&lt;ref name=&quot;:20&quot; /&gt;

Amnesty International described the law as a threat to civil liberties and warned that restrictions on filming police could weaken transparency and accountability.&lt;ref name=&quot;:21&quot; /&gt; The Committee to Protect Journalists (CPJ) also raised concerns that the law could deter media coverage of demonstrations and police activity.&lt;ref&gt;{{Cite web|url=https://cpj.org/2015/05/why-spains-new-gag-law-is-threat-to-free-flow-of-i/|title=Why Spain's new gag law is threat to free flow of information|last=Blogger|first=Borja Bergareche/CPJ Guest|date=2015-05-01|website=Committee to Protect Journalists|language=en-US|access-date=2026-03-02}}&lt;/ref&gt; In addition, United Nations Special Rapporteurs expressed concern that the law’s provisions were overly broad and risked arbitrary enforcement against peaceful protesters.&lt;ref&gt;{{Cite web|url=https://www.ohchr.org/en/press-releases/2015/02/two-legal-reform-projects-undermine-rights-assembly-and-expression-spain-un|title=“Two legal reform projects undermine the rights of assembly and expression in Spain” - UN experts|website=OHCHR|language=en|access-date=2026-03-02}}&lt;/ref&gt;

===== Javier Bauluz Case =====
One widely cited case involved Spanish photojournalist Javier Bauluz, a Pulitzer Prize–winning photographer, who was fined €960 under the Citizen Security Law after a confrontation with police while documenting migrant arrivals in the Canary Islands in November 2020.&lt;ref name=&quot;:22&quot; /&gt;&lt;ref name=&quot;:35&quot;&gt;{{Cite web|url=https://www.mfrr.eu/spain-fine-against-photographer-underscores-urgent-need-for-reform-of-gag-law/|title=Spain: Fine against photographer underscores urgent need for reform of Gag Law|last=MFRR|date=2022-06-21|website=Media Freedom Rapid Response|language=en-GB|access-date=2026-03-02}}&lt;/ref&gt; He had been photographing rescue boats arriving in Arguineguín, where thousands of migrants were being held in conditions later described by a judge as “deplorable.”&lt;ref name=&quot;:22&quot; /&gt; Video of the incident shows officers grabbing him and ordering him to leave, and he was later fined for “disrespecting an agent” and “refusing to identify himself,” though he said he had complied and was simply doing his job.&lt;ref name=&quot;:22&quot;&gt;{{Cite news|url=https://www.theguardian.com/world/2022/jun/14/photographer-capturing-migrant-camp-fined-1000-under-spains-gag-law|title=Photographer capturing migrant camp fined €1,000 under Spain’s ‘gag law’|last=Kassam|first=Ashifa|date=2022-06-14|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}&lt;/ref&gt;&lt;ref name=&quot;:35&quot; /&gt;

The fine arrived more than a year later and gave little explanation beyond citing provisions of the law. Bauluz rejected the sanction, arguing that police were limiting press access to prevent journalists from properly documenting the situation.&lt;ref name=&quot;:22&quot; /&gt;&lt;ref name=&quot;:35&quot; /&gt; He criticized the Gag Law for converting disputes into administrative fines imposed directly by authorities rather than matters handled through criminal courts.&lt;ref name=&quot;:22&quot; /&gt;

The case became a symbol of broader concerns that the law can be used to penalize journalists reporting on police activity. Press freedomorganizations and media groups condemned the fine and called for reform, arguing that the law enables arbitrary sanctions and threatens freedom of expression.&lt;ref name=&quot;:22&quot; /&gt;&lt;ref name=&quot;:35&quot; /&gt; Although Spain’s Constitutional Court upheld most of the law in 2021, critics continue to argue that reform is necessary to bring it in line with international human rights standards.&lt;ref name=&quot;:35&quot; /&gt;

===== Constitutional Court Review =====
Spain’s Constitutional Court reviewed the Citizen Security Law following multiple constitutional challenges. In Constitutional Court decision STC 172/2020, the Court upheld most provisions of the law but clarified limits on its application, particularly regarding sanctions for the use or dissemination of images of police officers. The Court emphasized that penalties cannot be applied in ways that effectively restrict legitimate journalistic reporting or public documentation of police activity.&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:37&quot;&gt;{{Cite web|url=https://hj.tribunalconstitucional.es/HJ/es/Resolucion/Show/26498|title=Sistema HJ - Resolución: SENTENCIA 172/2020|website=hj.tribunalconstitucional.es|access-date=2026-03-05}}&lt;/ref&gt; One of the most controversial aspects of the ruling was the Court’s decision to uphold the provision allowing administrative fines when photographs or videos of police officers are published in ways that could endanger an officer’s safety or interfere with an ongoing operation.

The Constitutional Court clarified that the mere act of recording or photographing police officers during public events or demonstrations is not automatically illegal. Instead, sanctions may only be imposed when the dissemination of those images creates a concrete risk to the safety of officers or interferes with a police operation.&lt;ref name=&quot;:37&quot; /&gt; For example, publishing images that reveal the identity of undercover officers or expose the location of police units during an active operation could justify sanctions. By contrast, photographing police activity during public demonstrations for journalistic reporting or public accountability generally falls within the protections of freedom of expression.&lt;ref name=&quot;:37&quot; /&gt;&lt;ref&gt;{{Cite web|url=https://solermartinabogados.com/en/can-i-record-the-police-can-they-force-me-to-erase-the-images-i-have-recorded-of-them/|title=Can I record the police in Spain? rights, limits|date=2025-10-06|language=en-US|access-date=2026-03-05}}&lt;/ref&gt; 

The Court emphasized that enforcement must respect constitutional guarantees of freedom of expression and assembly. However, it did not invalidate the core structure of the law, leaving its administrative sanction framework intact.&lt;ref name=&quot;:23&quot;&gt;{{Cite web|url=https://www.article19.org/resources/spain-time-to-end-to-repressive-gag-law/|title=Spain: Time to end repressive 'Gag Law'|date=2024-08-20|website=ARTICLE 19|language=en-US|access-date=2026-03-02}}&lt;/ref&gt;

===== The Ongoing Debate =====
The Citizen Security Law remains one of the most politically divisive laws in Spain’s contemporary democracy.

Supporters argue that the law provides necessary tools to maintain order and protect both police officers and the public. They stress that fines are administrative rather than criminal penalties and are subject to judicial review. From this perspective, the law regulates conduct rather than suppressing political ideas.

Critics, by contrast, argue that the law creates a climate of deterrence. Even without criminal prosecution, the risk of substantial fines may discourage citizens from participating in spontaneous demonstrations or from documenting police actions. Civil liberties groups describe this as a “chilling effect” on democratic participation.&lt;ref name=&quot;:19&quot; /&gt;&lt;ref name=&quot;:20&quot; /&gt;&lt;ref name=&quot;:21&quot; /&gt;

Reform efforts have repeatedly emerged in Spain’s national legislature, particularly from left-leaning parties that argue the law should be revised or partially repealed.&lt;ref name=&quot;:38&quot;&gt;{{Cite web|url=https://www.barrons.com/news/reform-of-spain-s-contested-security-law-fails-9b1f9a5|title=Reform Of Spain's Contested Security Law Fails|last=Presse|first=AFP-Agence France|website=barrons|language=en-us|access-date=2026-03-05}}&lt;/ref&gt; These parties contend that provisions related to protest, public demonstrations, and the recording of police activity give authorities too much discretion and risk discouraging political participation. By contrast, many right-leaning parties have defended the law, arguing that it provides necessary tools for maintaining public order and protecting police officers, especially during large demonstrations and periods of political unrest. As a result, proposals to substantially reform the law have often stalled due to political disagreement in parliament.&lt;ref name=&quot;:38&quot; /&gt;&lt;ref&gt;{{Cite web|url=https://monitor.civicus.org/explore/csos-warn-decision-not-to-reform-gag-law-is-bad-news-for-human-rights-in-spain/|title=CSOs warn decision not to reform “Gag Law” is “bad news for human rights in Spain”|website=Civicus Monitor|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://russpain.com/en/news-3/spanish-parliament-stuck-on-security-law-reform-398037/|title=The political scene is heating up: growing disagreements, unexpected pressure and intrigue in parliament|last=Rubio|first=Ricardo|date=2026-02-23|website=RUSSPAIN.COM|language=en-US|access-date=2026-03-05}}&lt;/ref&gt;

This divide reflects broader political tensions in Spain. Supporters of reform frequently frame the law as a legacy of a more security-focused approach to governance that emerged during periods of economic crisis and protest movements in the 2010s. Opponents of reform argue that weakening the law could undermine the ability of authorities to manage demonstrations and maintain public safety. Because these disagreements map closely onto Spain’s left-right political divide, efforts to significantly change the Citizen Security Law have proven difficult despite ongoing public debate.

== &lt;big&gt;Spain’s Historical Memory Act&lt;/big&gt; ==

===== Historical Background and Democratic Transition =====
[[File:Francisco Franco 1930.jpg|thumb|'''Francisco Franco in 1930, when he was still a rising officer in the Spanish army, years before the Spanish Civil War brought him to power and led to his long dictatorship.''']]
Spain’s contemporary debate over historical memory is rooted in the Spanish Civil War (1936–1939) and the subsequent dictatorship of General Francisco Franco, which lasted until 1975.&lt;ref&gt;{{Cite journal|last=Owens|first=Lawrence S.|date=2021|title=Timoteo Mendieta Alcalá and the Pact of Forgetting: trauma analysis of execution victims from a Spanish Civil War mass burial site at Guadalajara, Castilla la Mancha|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC8212665/|journal=Forensic Science International. Synergy|volume=3|pages=100156|doi=10.1016/j.fsisyn.2021.100156|issn=2589-871X|pmc=8212665|pmid=34179739}}&lt;/ref&gt; The war divided the country along political, ideological, and religious lines and resulted in widespread repression, imprisonment, and executions.&lt;ref&gt;{{Cite journal|last=Owens|first=Lawrence S.|date=2021|title=Timoteo Mendieta Alcalá and the Pact of Forgetting: trauma analysis of execution victims from a Spanish Civil War mass burial site at Guadalajara, Castilla la Mancha|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC8212665/|journal=Forensic Science International. Synergy|volume=3|pages=100156|doi=10.1016/j.fsisyn.2021.100156|issn=2589-871X|pmc=8212665|pmid=34179739}}&lt;/ref&gt;&lt;ref name=&quot;:24&quot;&gt;{{Cite journal|last=Boyd|first=Carolyn P.|date=2008|title=The Politics of History and Memory in Democratic Spain|url=https://www.jstor.org/stable/25098018|journal=The Annals of the American Academy of Political and Social Science|volume=617|pages=133–148|issn=0002-7162}}&lt;/ref&gt; After Franco’s victory, the regime promoted an official narrative that framed the conflict as a defense of national unity and Catholic identity.&lt;ref name=&quot;:24&quot; /&gt; Public monuments, street names, memorials, and religious symbols commemorating the dictatorship were erected throughout Spain.&lt;ref&gt;{{Cite web|url=https://dash.harvard.edu/server/api/core/bitstreams/e113f9f5-d512-4bfa-bd72-9c150cec2d32/content|title=Historical Memory in Post-Franco
Spain: Remembering a Purposely
Forgotten Past through
Memorialization at the Valle de los
Caídos in Cuelgamuros}}&lt;/ref&gt;

Following Franco’s death in 1975, Spain transitioned to democracy through a negotiated political process often referred to as the “Transition.”&lt;ref&gt;{{Cite web|url=https://api.drum.lib.umd.edu/server/api/core/bitstreams/cda590b3-0ba4-45b8-98c4-4333e42f5ed6/content|title=MEMORY AND RECONCILIATION IN THE
SPANISH TRANSITION TO DEMOCRACY:
1975-1982}}&lt;/ref&gt; During this period, political leaders adopted what became known as the “Pact of Forgetting” (Pacto del Olvido), an informal political understanding that prioritized reconciliation and democratic stability over reopening Civil War-era grievances.&lt;ref name=&quot;:25&quot;&gt;{{Cite web|url=https://journals.sagepub.com/action/cookieAbsent|title=Sage Journals: Discover world-class research|website=Sage Journals|language=en|doi=10.1177/026569149702700303|access-date=2026-03-02}}&lt;/ref&gt; The 1977 Amnesty Law granted broad amnesty for politically motivated crimes committed during the dictatorship.

By the early 2000s, civil society organizations began advocating for greater recognition of victims of Franco-era repression, including efforts to identify mass graves and remove public symbols associated with the dictatorship.&lt;ref&gt;{{Cite web|url=https://scispace.com/pdf/the-return-of-civil-war-ghosts-the-ethnography-of-2j8mponmed.pdf|title=The return of Civil War ghosts
The ethnography of exhumations in contemporary Spain}}&lt;/ref&gt; Supporters argued that democratic consolidation required public acknowledgment of historical injustices.&lt;ref&gt;{{Cite web|url=https://www.researchgate.net/publication/254084329_Determinants_of_Attitudes_Toward_Transitional_Justice_An_Empirical_Analysis_of_the_Spanish_Case|title=Determinants of Attitudes Toward Transitional Justice: An Empirical Analysis of the Spanish Case}}&lt;/ref&gt; In response, Spain enacted Law 52/2007, commonly known as the Historical Memory Act.&lt;ref name=&quot;:26&quot;&gt;{{Cite web|url=https://reparations.qub.ac.uk/assets/uploads/Ley-52-2007-Spain-EN.pdf|title=Ley 52-2007 Spain EN.docx}}&lt;/ref&gt;

===== The 2007 Historical Memory Act =====
Law 52/2007 recognizes and expands rights for individuals who suffered persecution or violence during the Civil War and dictatorship. Its preamble states that it is not the role of the legislator to impose a specific collective memory, but rather to promote democratic values and protect personal and family memory as expressions of democratic citizenship.&lt;ref name=&quot;:26&quot; /&gt;

At the same time, the law mandates the removal of “shields, insignia, plaques and other objects or commemorative mentions” that exalt the military uprising, Civil War, or repression of the dictatorship from public buildings and spaces.&lt;ref name=&quot;:26&quot; /&gt; It also supports efforts to locate and identify victims of repression and provides symbolic recognition to those who suffered under the regime.

The Act represents a shift from the earlier policy of institutional silence toward a more active engagement with the legacy of the dictatorship.&lt;ref name=&quot;:25&quot; /&gt;&lt;ref name=&quot;:26&quot; /&gt;

The main provisions are:
* Official recognition of victims of political, religious, and ideological violence on both sides of the Civil War and under Franco’s rule
* Formal condemnation of the Franco regime
* Ban on political events at the Valley of the Fallen, where Franco was buried
* Removal of public symbols, plaques, statues, and insignia that celebrate the military coup or the dictatorship (with limited exceptions for artistic, architectural, or religious reasons)
* Government support for locating, identifying, and exhuming victims buried in mass graves
* Spanish citizenship granted to surviving members of the International Brigades without requiring them to give up their original nationality
* Declaration that Franco-era trials and laws lacked legitimacy
* Temporary changes to nationality rules allowing people who left Spain during the dictatorship — and their descendants — to reclaim Spanish citizenship
* Financial and symbolic assistance for victims and their families

===== Supporters’ and Critics’ Perspective =====
Supporters of the Historical Memory Act argue that it reflects a mature democratic commitment to historical justice and human dignity.&lt;ref name=&quot;:27&quot;&gt;{{Cite news|url=https://www.theguardian.com/world/2022/oct/05/spain-passes-law-to-bring-dignity-to-franco-era-victims|title=Spain passes law to bring ‘justice’ to Franco-era victims|last=Jones|first=Sam|date=2022-10-05|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}&lt;/ref&gt;&lt;ref name=&quot;:28&quot;&gt;{{Cite news|url=https://www.nytimes.com/2007/10/28/world/europe/28spain.html|title=Bill in Spanish Parliament Aims to End ‘Amnesia’ About Civil War Victims|last=Burnett|first=Victoria|date=2007-10-28|work=The New York Times|access-date=2026-03-02|language=en-US|issn=0362-4331}}&lt;/ref&gt; From this perspective, a constitutional democracy cannot maintain public symbols that glorify authoritarian rule. Removing such symbols is seen not as erasing history, but as ending official state endorsement of a particular political narrative.&lt;ref name=&quot;:28&quot; /&gt;

Advocates also emphasize the “right to truth” for victims and their families, aligning Spain with broader international human rights standards concerning recognition, memory, and accountability.&lt;ref&gt;{{Cite web|url=https://www.swisspeace.ch/assets/publications/downloads/Gonzalez-Garcia_WorkingPaper_2_2023.pdf|title=The Search for Truth in Spain: Debates Around the Creation of a Truth Commission}}&lt;/ref&gt; Reports by United Nations Special Rapporteurs have encouraged Spain to strengthen efforts related to truth, justice, and reparation for victims of Franco-era repression.&lt;ref&gt;{{Cite web|url=https://news.un.org/en/story/2014/02/461222|title=UN expert urges Spain to probe alleged atrocities during 1930's civil war}}&lt;/ref&gt;

For supporters, the law corrects decades of imbalance in public memory and promotes constitutional values grounded in democracy and human rights.&lt;ref name=&quot;:27&quot; /&gt;&lt;ref&gt;{{Cite web|url=https://www.amnesty.org/en/documents/eur41/001/2013/en/|title=Spain: Supreme Court overturns ban on full-face veils; AI concerns remain about restrictions on headscarves in schools|date=2013-04-08|website=Amnesty International|language=en|access-date=2026-03-02}}&lt;/ref&gt;

Critics argue that the Historical Memory Act risks politicizing historical interpretation by privileging one narrative over others.&lt;ref name=&quot;:29&quot;&gt;{{Cite news|url=https://www.economist.com/europe/2020/09/17/the-spanish-government-proposes-a-new-law-on-history|title=The Spanish government proposes a new law on history|work=The Economist|access-date=2026-03-02|issn=0013-0613}}&lt;/ref&gt;&lt;ref name=&quot;:30&quot;&gt;{{Cite news|url=https://www.nytimes.com/2007/10/24/world/europe/24iht-spain.4.8039804.html|title=Spain undergoes wrenching awakening from 'amnesia'|last=Burnett|first=Victoria|date=2007-10-24|work=The New York Times|access-date=2026-03-02|language=en-US|issn=0362-4331}}&lt;/ref&gt; Some scholars contend that legislative intervention in historical memory can transform contested historical debate into state-defined orthodoxy.&lt;ref name=&quot;:30&quot; /&gt; Opponents also argue that removing monuments may constitute symbolic erasure rather than genuine reconciliation.&lt;ref name=&quot;:29&quot; /&gt; They maintain that democratic societies should allow historical interpretation to evolve through open public discourse rather than through statutory mandates.&lt;ref name=&quot;:30&quot; /&gt;

Much of this debate has centered on the Valle de los Caídos (Valley of the Fallen) memorial complex, one of the most prominent and controversial symbols associated with Spain’s Civil War and the Franco dictatorship. The massive monument, built after the war and located near Madrid, contains a basilica carved into a mountain and a large cross that dominates the surrounding landscape. For decades it served as the burial site of General Francisco Franco as well as thousands of victims from both sides of the Civil War.&lt;ref&gt;{{Cite web|url=https://www.bbc.com/news/world-europe-50164806|title=Franco exhumation: Spanish dictator's remains moved|date=2019-10-24|website=www.bbc.com|language=en-GB|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref&gt;{{Cite news|url=https://www.theguardian.com/world/2019/oct/24/franco-exhumation-spain-dictator-madrid|title='Spain is fulfilling its duty to itself': Franco's remains exhumed|last=Jones|first=Sam|date=2019-10-24|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}&lt;/ref&gt; Supporters of Spain’s memory laws argue that the site symbolized the continued public prominence of Franco’s regime, while critics argue that the complex represents an important historical monument whose meaning should be debated rather than reshaped through legislation.

The controversy intensified when the Spanish government ordered the exhumation of Franco’s remains from the site in 2019, relocating them to a different cemetery.&lt;ref&gt;{{Cite web|url=https://www.lamoncloa.gob.es/lang/en/presidente/news/Paginas/2019/20191024-statement.aspx|title=Institutional statement by Acting President of the Government regarding exhumation of Francisco Franco}}&lt;/ref&gt; The government justified the decision as part of a broader democratic memory policy aimed at preventing the memorial from functioning as a place of political homage to the dictatorship. Critics, however, viewed the move as politically motivated and reflective of Spain’s continuing polarization over how the country should confront its past.

===== Ongoing Debate: Truth, Memory, and Democratic Pluralism =====
Spain’s memory laws have become one of the most visible and contested areas of contemporary public debate. The discussion centers on how a democracy should address a painful past and what role the state should play in shaping public memory. In Spain, this debate appears in disputes over monuments, commemorations, public spaces, and the official recognition of victims of the Civil War and Franco’s dictatorship.&lt;ref name=&quot;:31&quot;&gt;{{Cite web|url=https://www.theartnewspaper.com/2023/06/30/debate-rages-in-spain-over-how-to-rememberor-forgetfranco-dictatorship|title=Debate rages in Spain over how to remember—or forget—Franco's dictatorship|last=Coego|first=Alexandra F.|date=2023-06-30|website=The Art Newspaper - International art news and events|language=en|access-date=2026-03-02}}&lt;/ref&gt;&lt;ref name=&quot;:32&quot;&gt;{{Cite web|url=https://jacobin.com/2024/01/spain-memory-law-ghosts-francoism|title=Spain’s Memory Law Hasn’t Banished the Ghosts of Francoism|last=By|website=jacobin.com|language=en-US|access-date=2026-03-02}}&lt;/ref&gt;

Supporters of the Democratic Memory framework argue that removing Francoist symbols and formally recognizing victims strengthens democracy. They maintain that a constitutional state should not continue to honor an authoritarian regime in public spaces. From this perspective, memory laws do not erase history but instead end state endorsement of dictatorship and affirm the dignity of those who suffered repression.&lt;ref name=&quot;:31&quot; /&gt;&lt;ref name=&quot;:33&quot;&gt;{{Cite web|url=https://www.reuters.com/world/spain-pays-tribute-francos-victims-50-years-after-his-death-2025-10-31/|title=Spain pays tribute to Franco's victims 50 years after his death}}&lt;/ref&gt;

Critics argue that legislating memory can deepen political divisions. Some commentators warn that when the government takes an active role in defining historical meaning, it risks turning complex historical debates into partisan conflicts.&lt;ref name=&quot;:32&quot; /&gt;&lt;ref&gt;{{Cite web|url=https://jacobin.com/2024/01/spain-memory-law-ghosts-francoism|title=Spain’s Memory Law Hasn’t Banished the Ghosts of Francoism|last=By|website=jacobin.com|language=en-US|access-date=2026-03-02}}&lt;/ref&gt; Articles examining Spain’s evolving memory laws describe a society still divided over how to interpret the Civil War and Franco’s legacy, with disagreement over whether these reforms promote justice or contribute to polarization.&lt;ref&gt;{{Cite web|url=https://enrs.eu/article/spanish-controversies-related-to-memory|title=Spanish controversies related to memory|website=ENRS|language=en|access-date=2026-03-02}}&lt;/ref&gt;

In today’s Spain, historical memory is not only about the past. It remains tied to ongoing debates about national identity, democracy, and constitutional values.&lt;ref name=&quot;:32&quot; /&gt;&lt;ref name=&quot;:33&quot; /&gt; The regulation of collective memory shows how law, history, and public expression intersect in a modern democratic society.

== &lt;big&gt;Religious Freedom in Spain&lt;/big&gt; ==

===== Historical Development =====
Spain’s religious history is defined less by steady liberalization than by recurring struggles over whether religious belief could appear in public at all. Medieval coexistence among Christians, Muslims, and Jews existed, but it never displaced the stronger political impulse toward religious unity enforced through law.&lt;ref name=&quot;:39&quot;&gt;{{Cite journal|last=Montserrat|first=Daniel B.|date=1995|title=The Constitutional Development of Religious Freedom in Spain: An Historical Analysis|url=https://ir.law.fsu.edu/cgi/viewcontent.cgi?article=1241&amp;context=jtlp|journal=Fla. St. U. J. Transnat’l L. &amp; Pol’y|volume=4|pages=27}}&lt;/ref&gt;

During the early constitutional period that impulse was embedded directly into state structures. The Constitution of Cádiz (1812) combined political liberalism with explicit Catholic exclusivity, requiring public officials to swear to defend Catholicism and mandating religious instruction in schools.&lt;ref name=&quot;:39&quot; /&gt; Religion was not only protected; it was communicated through state institutions.

Later constitutions softened these rules but continued to restrict public expression. Non-Catholic worship was sometimes tolerated, but often confined to private settings, allowing belief without visible organization or expression.&lt;ref name=&quot;:39&quot; /&gt;

The Spanish Constitution of 1931 marked a sharp shift by restricting the Catholic Church’s institutional role, removing funding, dissolving religious orders, and limiting religious education.&lt;ref name=&quot;:40&quot;&gt;{{Cite journal|last=Combalia|first=Zolia|last2=Roca|first2=Maria|date=2010|title=Religion and the Secular State of Spain, in Religion and the Secular State|url=https://original.religlaw.org/content/blurb/files/Spain%202014.pdf|journal=(W. Cole Durham, Jr. &amp; Javier Martínez-Torrón eds., 2015)|pages=661}}&lt;/ref&gt; Rather than establishing neutrality, this reallocated control over how religion could appear in public institutions. This approach reflected a broader European trend during the early twentieth century, as communist and strongly secular regimes sought to remove religion from public life altogether. In those systems, religious expression was not merely regulated but suppressed—public worship, teaching, and institutional presence were restricted or eliminated and replaced with state-controlled ideological messaging. Spain’s 1931 model did not go as far, but it operated within the same broader movement toward limiting religion’s visibility in public communication and institutional life.&lt;ref name=&quot;:40&quot; /&gt;

Under Franco, Catholicism was restored as the central organizing force of public life. Religious teaching, symbols, and institutional presence were again integrated into education and law, but limited almost entirely to a single faith.&lt;ref name=&quot;:39&quot; /&gt;&lt;ref name=&quot;:40&quot; /&gt;

The 1978 Constitution breaks from both models. Article 16 provides:

* “Freedom of ideology, religion and worship of individuals and communities is guaranteed… No one may be compelled to declare his ideology, religion or beliefs… No religion shall have a state character….”&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1978-31229|title=Constitución Española art. 16 (1978), BOE-A-1978-31229|website=www.boe.es|access-date=2026-04-20}}&lt;/ref&gt;

This provision protects both private belief and public expression, prevents compelled disclosure, and removes any official state religion, while still allowing cooperation with religious groups. Religion remains visible in public life, but it is no longer directed by the state.

===== Modern Spain: Public Expression, Institutional Practice, and European Context =====
Spain’s modern framework allows religion to operate openly in public life while maintaining formal neutrality. Religious expression appears in education and public institutions as individual and community activity protected by law rather than as state endorsement.&lt;ref name=&quot;:40&quot; /&gt;

This includes the ability to access religious instruction in public schools when requested and to organize religious communities with legal recognition.&lt;ref name=&quot;:40&quot; /&gt; These practices reflect a system where religious expression remains visible in ordinary public settings rather than confined to private belief.

This approach aligns with international human rights standards. The Spanish Constitution prohibits discrimination based on religion and guarantees the right to practice religion publicly or privately, consistent with broader protections of religious expression.&lt;ref name=&quot;:41&quot;&gt;{{Cite web|url=https://www.state.gov/reports/2023-report-on-international-religious-freedom/spain/|title=U.S. Dep’t of State, 2023 Report on International Religious Freedom: Spain (May 1, 2024)|website=United States Department of State|language=en-US|access-date=2026-04-20}}&lt;/ref&gt;

Spain operates within a broader European human rights framework that applies the margin of appreciation doctrine, allowing individual countries to adopt different approaches to religion in public life.&lt;ref&gt;{{Cite journal|last=Lugato|first=Monica|date=2013|title=The “Margin of Appreciation” and Freedom of Religion|url=https://scholarship.law.stjohns.edu/cgi/viewcontent.cgi?article=1061&amp;context=jcls|journal=J. Cath. Legal Stud.|volume=52|pages=49}}&lt;/ref&gt;&lt;ref name=&quot;:48&quot;&gt;{{Cite journal|last=Rayón Ballesteros|first=María Concepción|date=2025|title=Generative AI for Lawyers in Spain: A balanced
approach to the legal framework, technical foundations
and best practices, combining technological innovation
with professional responsibility|url=https://reference-global.com/article/10.2478/law-2025-0003|journal=Complutense University, Law and Business|volume=5|pages=12}}&lt;/ref&gt; Under this doctrine, the European Court of Human Rights permits states to balance religious expression and public order according to their own legal traditions.&lt;ref name=&quot;:48&quot; /&gt;

The contrast with France illustrates this flexibility. France restricts visible religious symbols such as hijabs or large crosses in public schools under Law No. 2004-228 of 15 March 2004,¹ a policy upheld by the European Court of Human Rights in ''Jasvir Singh v. France''.&lt;ref&gt;{{Cite web|url=https://www.legifrance.gouv.fr/jorf/id/JORFTEXT000000417977|title=Law No. 2004-228 of March 15, 2004, regulating, in application of the principle of secularism, the wearing of signs or dress manifesting a religious affiliation in public primary and secondary schools, JOURNAL OFFICIEL DE LA RÉPUBLIQUE FRANÇAISE [J.O.] [OFFICIAL GAZETTE OF FRANCE], Mar. 17, 2004, p. 5190.}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22002-1403%22%5D%7D|title=Singh v. France, App. No. 48321/08, Eur. Ct. H.R. (2009).|website=hudoc.echr.coe.int|access-date=2026-05-03}}&lt;/ref&gt; In Spain, by contrast, similar forms of expression are generally permitted, and wearing religious symbols is treated as an individual act rather than a violation of neutrality.

Spain’s system also allows religious institutions to participate directly in public education. In Fernández Martínez v. Spain, the European Court of Human Rights upheld Spain’s ability to allow the Catholic Church to control who teaches Catholic religion in public schools.&lt;ref name=&quot;:42&quot;&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng?i=001-145068|title=Fernández Martínez v. Spain, App. No. 56030/07, Eur. Ct. H.R. (Grand Chamber June 12, 2014)|website=hudoc.echr.coe.int|access-date=2026-04-20}}&lt;/ref&gt; The case involved a teacher who lost his position after publicly opposing Church teachings. The Court accepted that religious institutions may define who represents their message in educational settings.

Despite formal equality, differences remain in practice, and they are best understood as largely natural rather than artificial. Spain’s legal framework is neutral, but historical and demographic factors shape how religious expression appears. For example, Catholic religious instruction is more widely available in public schools because Catholicism has a larger institutional presence and more students requesting it, not because the law excludes other faiths.&lt;ref name=&quot;:40&quot; /&gt;&lt;ref name=&quot;:42&quot; /&gt; Other groups have the same legal rights but less visible participation due to size and infrastructure.

The result is a system in which religion remains active and visible in public life without formal state endorsement. Spain does not remove religion from public space; it regulates how it appears and ensures that participation remains voluntary.

== &lt;big&gt;The Right to Be Forgotten&lt;/big&gt; ==

===== Google Spain and the Transition to the General Data Protection Regulation =====
A seminal case in modern data protection law arose from Spain and reshaped the relationship between privacy and access to information in the digital age. In ''Google Spain SL v. AEPD and Mario Costeja González'', Spain’s Audiencia Nacional asked whether EU data protection law could require a search engine to remove links to lawful, truthful information appearing in name-based searches.&lt;ref name=&quot;:43&quot;&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:62012CJ0131|title=Google Spain SL v. Agencia Española de Protección de Datos (AEPD), Case C-131/12 (CJEU 2014).|website=eur-lex.europa.eu|access-date=2026-04-20}}&lt;/ref&gt;

The dispute stemmed from a 1998 notice in ''La Vanguardia'' announcing the forced sale of Mario Costeja González’s property for unpaid social security debts. Although the publication was lawful, its later digitization made it easily accessible through search engines, effectively reviving a long-resolved matter. Costeja requested removal of the links, and the Spanish Data Protection Agency ordered Google to de-list them while allowing the newspaper to remain online.&lt;ref name=&quot;:43&quot; /&gt;

The Court of Justice of the European Union held that individuals may request removal of links where the information is “inadequate, irrelevant or no longer relevant,” even if the original publication remains lawful.&lt;ref name=&quot;:43&quot; /&gt; It reasoned that search engines act as “data controllers” because they organize and present personal data in a way that significantly affects privacy. The legal harm, therefore, arises not from the original publication, but from the amplified visibility created by search engines.&lt;ref name=&quot;:44&quot;&gt;{{Cite journal|last=Post|first=Robert C.|date=2017|title=Data Privacy and Dignitary Privacy: Google Spain, the Right to Be Forgotten, and the Construction of the Public Sphere|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2953468|journal=Yale Law School, Public Law Research Paper|volume=598}}&lt;/ref&gt;

Although ''Google Spain'' was decided under Directive 95/46/EC, that framework has since been repealed and replaced by the General Data Protection Regulation (GDPR).&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/1995/46/oj/eng|title=Directive 95/46/EC arts. 2(a), 2(b).|website=eur-lex.europa.eu|access-date=2026-04-20}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://gdpr-info.eu/|title=General Data Protection Regulation (GDPR) – Legal Text|website=General Data Protection Regulation (GDPR)|language=en-US|access-date=2026-04-30}}&lt;/ref&gt; The repeal did not eliminate the right recognized in the case. Instead, the principle was codified and strengthened in Article 17 of the GDPR, which establishes a “right to erasure.” Article 17 allows individuals to request deletion of personal data that is no longer necessary, relevant, or lawfully processed, and requires controllers, where feasible, to take reasonable steps to inform other entities processing that data, extending the effect of erasure beyond a single source.&lt;ref name=&quot;:45&quot;&gt;{{Cite web|url=https://gdpr-info.eu/art-17-gdpr/|title=Art. 17 GDPR – Right to erasure (‘right to be forgotten’)|website=General Data Protection Regulation (GDPR)|language=en-US|access-date=2026-04-30}}&lt;/ref&gt;

Crucially, Article 17 is grounded in Articles 7 and 8 of the Charter of Fundamental Rights of the European Union. Article 7 guarantees the right to respect for private and family life, while Article 8 establishes a distinct right to the protection of personal data, requiring that such data be processed fairly and subject to independent oversight.&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/eu-charter/title/title-ii-freedoms|title=Charter of Fundamental Rights of the European Union arts. 7–8.|website=fra.europa.eu|access-date=2026-04-30}}&lt;/ref&gt; In ''Google Spain'', these provisions justified the Court’s conclusion that search results displaying outdated personal information can constitute an ongoing interference with private life and that search engines, as data controllers, must respond to requests for removal.&lt;ref name=&quot;:43&quot; /&gt;

Unlike the Directive, the GDPR applies directly across all Member States, creating a more uniform and enforceable framework. EU law further provides that references to the repealed Directive are to be read as references to the GDPR, preserving continuity between ''Google Spain'' and the current legal regime.&lt;ref name=&quot;:45&quot; /&gt;

===== The Modern Doctrine of the Right to Be Forgotten =====
While ''Google Spain'' established the right to be forgotten, subsequent case law has transformed it into a structured doctrine grounded in Article 17 of the GDPR and the Charter. The modern right is not a mechanism to erase the past, but a balancing framework that evaluates whether continued access to personal information remains justified in light of both privacy and communication interests.

Under Article 17, individuals may request erasure where data is no longer necessary, is inaccurate, or is unlawfully processed, but the right is not absolute.&lt;ref name=&quot;:45&quot; /&gt; Courts apply a case-by-case balancing test rooted in Articles 7 and 8, weighing the individual’s privacy and data protection rights against the public’s interest in access to information. In practice, this inquiry turns on factors such as accuracy, passage of time, the individual’s role in public life, and whether the information contributes to a matter of legitimate public concern. This balancing directly implicates core communication law values, including freedom of expression, the public’s right to receive information, and the preservation of an accurate public record.

In ''Google LLC v. CNIL'', the Court addressed the geographic scope of the doctrine.&lt;ref name=&quot;:46&quot;&gt;{{Cite web|url=https://infocuria.curia.europa.eu/tabs/document?source=document&amp;docid=218105&amp;doclang=EN|title=Google v. CNIL, Case C-507/17 (CJEU 2019).|website=infocuria.curia.europa.eu|access-date=2026-04-20}}&lt;/ref&gt; The French data protection authority mandated global delisting, arguing that limiting content removal to European domains rendered the &quot;right to be forgotten&quot; ineffective. They asserted that removing search results only on regional sites (e.g., google.fr) allowed users to easily bypass the restriction by accessing global versions (e.g., google.com). The Court rejected that position, holding that EU law does not require global de-referencing. Instead, search engines are required to ensure effective content removal within the European Union by using geo-blocking to prevent access from non-EU domains. This approach limits the application of EU privacy laws to its own borders, avoiding global, extraterritorial censorship that would conflict with stronger international speech protections&lt;ref name=&quot;:46&quot; /&gt;

In ''GC and Others v. CNIL'', the Court refined the balancing test for sensitive categories of information.&lt;ref name=&quot;:51&quot;&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:62017CJ0136|title=GC &amp; Others v. Commission Nationale de l’Informatique et des Libertés (CNIL), Case C-136/17, ECLI:EU:C:2019:773 (Ct. Just. Eur. Union Sept. 24, 2019).|website=eur-lex.europa.eu|access-date=2026-04-30}}&lt;/ref&gt; The case involved requests to remove links containing data about political affiliations, religious beliefs, and criminal history. The Court held that such data requires heightened protection, but not automatic removal. Instead, search engines must determine whether continued access is “strictly necessary” for the public’s right to information.&lt;ref name=&quot;:51&quot; /&gt; For example, information about a politician’s past conduct may remain accessible because it informs democratic decision-making, while similar information about a private individual is more likely to be removed. This standard effectively requires search engines to evaluate whether speech contributes to public discourse, placing them in a quasi-adjudicative role traditionally occupied by courts.

In ''TU and RE v. Google LLC'', the Court addressed inaccurate or misleading information.&lt;ref name=&quot;:54&quot;&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:62020CJ0460|title=TU &amp; RE v. Google LLC, Case C-460/20, ECLI:EU:C:2022:962 (Ct. Just. Eur. Union Dec. 8, 2022).|website=eur-lex.europa.eu|access-date=2026-04-30}}&lt;/ref&gt; The applicants challenged articles criticizing their business practices, arguing that the information was false or distorted. The Court held that individuals need not first obtain a judicial ruling to prove falsity. Instead, if they provide relevant and sufficient evidence that the information is manifestly inaccurate, the search engine must delist it, including associated thumbnail images.&lt;ref name=&quot;:54&quot; /&gt; This ruling has significant implications for communication law, as it creates a mechanism similar to defamation law within data protection, allowing individuals to challenge harmful or misleading content without initiating formal litigation while requiring platforms to assess the accuracy of speech.

At the same time, the Court has emphasized limits grounded in the public interest. In ''Camera di Commercio di Lecce v. Manni'', the Court rejected a request to remove personal data from a public commercial register documenting a past bankruptcy.&lt;ref name=&quot;:55&quot;&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:62015CJ0398|title=Camera di Commercio, Industria, Artigianato e Agricoltura di Lecce v. Salvatore Manni, Case C-398/15, ECLI:EU:C:2017:197 (Ct. Just. Eur. Union Mar. 9, 2017).|website=eur-lex.europa.eu|access-date=2026-04-30}}&lt;/ref&gt; It held that such records serve the public interest in legal certainty, transparency, and market reliability.&lt;ref name=&quot;:55&quot; /&gt; This reflects a longstanding communication law principle: certain categories of information—particularly official records—retain enduring public value and cannot be erased simply because they are reputationally harmful.

Taken together, these cases show that the right to be forgotten is really about whether information should still be easy to find through a name search. If it no longer serves a real public purpose, it can be removed from search results; if it does, it stays. In practice, search engines make that call first, which means they end up deciding what information about a person remains visible online.

== &lt;big&gt;Spain in the AI Era&lt;/big&gt; ==

===== Artificial Intelligence and the Changing Nature of Communication =====
Artificial intelligence is reshaping a core assumption of communication law: that speech and images can be reliably traced to a real speaker. Technologies like deepfakes, AI-generated influencers, and algorithmic content systems now allow communication to circulate without a clear human source, raising new questions about attribution, truth, and accountability. Spain offers a useful lens for how legal systems are responding.

Spain does not regulate artificial intelligence through a single, unified statute. Instead, it operates within the framework of the European Union’s Artificial Intelligence Act, which establishes a risk-based regulatory model across Member States.&lt;ref name=&quot;:47&quot;&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2024/1689/oj/eng|title=Regulation (EU) 2024/1689 (Artificial Intelligence Act).|website=eur-lex.europa.eu|access-date=2026-04-20}}&lt;/ref&gt; High-risk systems, such as those used in employment or public decision-making, must comply with obligations including transparency, human oversight, and safeguards against bias.&lt;ref name=&quot;:47&quot; /&gt; Whether generative AI tools fall within these categories depends on how they are used and the extent to which they influence real-world or communicative outcomes.

At the national level, Spain is developing a Law on the Good Use and Governance of Artificial Intelligence, which will supplement the EU framework and introduce enforcement mechanisms, including the ability to suspend harmful AI systems.&lt;ref name=&quot;:49&quot;&gt;{{Cite web|url=https://www.cuatrecasas.com/es/spain/propiedad-intelectual/art/anteproyecto-ley-buen-uso-gobernanza-ia|title=Draft Law on the Good Use and Governance of Artificial Intelligence (Anteproyecto de Ley para el Buen Uso y la Gobernanza de la Inteligencia Artificial) (unpublished draft). (approved by Consejo de Ministros Mar. 11, 2025).|website=Cuatrecasas|language=es|access-date=2026-04-20}}&lt;/ref&gt; Spain has also created the Agencia Española de Supervisión de Inteligencia Artificial (AESIA) as its central regulator.&lt;ref&gt;Royal Decree 817/2023 of November 8, establishing a controlled testing environment for artificial intelligence systems (Real Decreto 817/2023, de 8 de noviembre), B.O.E. No. 269, Nov. 10, 2023 (Spain).&lt;/ref&gt; Notably, through Royal Decree 817/2023, Spain became the first country in the European Union to implement an AI regulatory sandbox, allowing high-risk systems to be tested under real-world conditions and effectively piloting compliance with the EU AI Act before full enforcement.&lt;ref&gt;General Audiovisual Communication Law 13/2022 of July 7 (Ley 13/2022, de 7 de julio, General de Comunicación Audiovisual), B.O.E. No. 163, July 8, 2022 (Spain).&lt;/ref&gt;

Spain does not yet have a standalone AI statute, but its existing legal framework, particularly in areas like media, privacy, and commercial regulation, already shapes how artificial intelligence operates in practice.

===== How Spain is Currently Regulating AI =====
Spain’s audiovisual and media laws are beginning to directly address AI-generated communication. Under Law 13/2022 on Audiovisual Communication, content creators, including high-level influencers, can be treated as audiovisual service providers and are responsible for the content they distribute.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2022-11311|title=BOE-A-2022-11311 Ley 13/2022, de 7 de julio, General de Comunicación Audiovisual.|website=www.boe.es|access-date=2026-04-30}}&lt;/ref&gt; This matters in the AI context because creators using AI-generated avatars, voices, or deepfake-style content for advertising must ensure that such material is not misleading. For example, an influencer using an AI-generated voice to promote a product without disclosure could face liability for deceptive commercial communication.

This emphasis on transparency is reinforced in Spain’s draft law the Good Use and Governance of Artificial Intelligence, which treats the undisclosed use of deepfakes as a serious regulatory violation in many contexts.&lt;ref name=&quot;:49&quot; /&gt; 

Reputation and dignity are also central concerns. Organic Law 1/1982 protects the right to honor, privacy, and self-image, and its application to AI-generated content is increasingly significant.&lt;ref name=&quot;:13&quot; /&gt; Disseminating a non-consensual deepfake, such as placing a person’s likeness into fabricated media, can constitute an illegitimate intromission into that person’s rights, even if the content is artificially generated. Spanish doctrine is increasingly moving toward what scholars describe as algorithmic honor: the idea that harm to reputation can arise from automated systems themselves, regardless of human intent. This aligns with Spanish Supreme Court jurisprudence recognizing that reputational harm caused by automated or data-driven systems may still trigger liability where the effect is injurious.&lt;ref&gt;{{Cite web|url=https://vlex.es/vid/922052595|title=See, e.g., Tribunal Supremo [T.S.] [Supreme Court], Judgment No. 35/2023 (Spain).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt;

Closely related is the right to correct false information. Organic Law 2/1984 establishes a traditional right of rectification, allowing individuals to demand correction of inaccurate public statements.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1984-7248|title=BOE-A-1984-7248 Ley Orgánica 2/1984, de 26 de marzo, reguladora del derecho de rectificación.|website=www.boe.es|access-date=2026-04-20}}&lt;/ref&gt; In the digital era, this concept is reinforced by Spain’s Organic Law 3/2018 on Data Protection and Digital Rights, which includes a modern digital rectification right requiring platforms to address inaccurate or misleading personal data.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2018-16673|title=BOE-A-2018-16673 Ley Orgánica 3/2018, de 5 de diciembre, de Protección de Datos Personales y garantía de los derechos digitales.|website=www.boe.es|access-date=2026-04-20}}&lt;/ref&gt; In practice, this provides a legal tool against AI-generated falsehoods, such as fabricated biographies or hallucinated statements, by requiring platforms or publishers to correct the record.

Spain’s approach also extends into advertising law. Under Law 3/1991 on Unfair Competition, commercial practices must not mislead consumers.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1991-628|title=BOE-A-1991-628 Ley 3/1991, de 10 de enero, de Competencia Desleal.|website=www.boe.es|access-date=2026-04-30}}&lt;/ref&gt; This applies directly to AI-generated endorsements or testimonials. For instance, if a company deploys an AI-generated persona that appears as specific person to promote a product without disclosure, regulators may treat this as deceptive advertising because it manipulates the audience’s trust in human communication.

At a structural level, Spain is also confronting the role of algorithms in shaping communication itself. The Rider Law, enacted through Royal Decree-Law 9/2021, requires companies to disclose the parameters and logic of algorithms that affect workers’ conditions.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2021-7840|title=BOE-A-2021-7840 Real Decreto-ley 9/2021, de 11 de mayo, por el que se modifica el texto refundido de la Ley del Estatuto de los Trabajadores, aprobado por el Real Decreto Legislativo 2/2015, de 23 de octubre, para garantizar los derechos laborales de las personas dedicadas al reparto en el ámbito de plataformas digitales.|website=www.boe.es|access-date=2026-04-20}}&lt;/ref&gt; While rooted in labor law, this requirement has clear communicative implications because it forces organizations to explain how algorithmic systems decide and communicate outcomes such as hiring, firing, or task allocation. Spanish courts have upheld this transparency obligation, confirming that algorithmic decision-making can be treated as a form of accountable communication within legal frameworks.&lt;ref&gt;{{Cite web|url=https://hj.tribunalconstitucional.es/en/Resolucion/Show/29590|title=Tribunal Constitucional [T.C.] [Constitutional Court], Judgment No. 80/2023 (Spain).|website=hj.tribunalconstitucional.es|access-date=2026-04-30}}&lt;/ref&gt;

This logic reaches its clearest expression in the Spanish Supreme Court’s BOSCO decision.&lt;ref&gt;{{Cite web|url=https://vlex.es/vid/1091644276|title=Tribunal Supremo [T.S.] [Supreme Court], Judgment No. 1119/2025 (Spain) (BOSCO case).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt; There, the Court required the government to disclose the logic of an automated system used to determine eligibility for public benefits. From a communication law perspective, the ruling treats algorithmic outputs as a form of state communication. If the government uses an automated system to speak to citizens through decisions, it must also explain that reasoning. Transparency thus becomes a constitutional requirement tied to the public’s right to information.

At the same time, BOSCO exposes a deeper constitutional tension. The right of access to information under Article 105(b) of the Spanish Constitution may conflict with protections for intellectual property and trade secrets under Article 33.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/pdf/1978/BOE-A-1978-40001-consolidado.pdf|title=Constitución Española, B.O.E. No. 311, Dec. 29, 1978, arts. 33, 105(b) (Spain).}}&lt;/ref&gt; The Court’s reasoning suggests that, at least where fundamental rights are implicated, public communicative accountability can outweigh private commercial secrecy. This marks a significant shift in how communication law interacts with technology, as the logic behind speech itself may become subject to disclosure.

Across these areas, a common theme emerges. Spain is not treating AI as a separate legal problem requiring entirely new doctrines. Instead, it is adapting existing communication law principles, including truthfulness, transparency, dignity, and accountability, to new technological conditions. The result is a framework in which AI-generated communication is regulated not by its novelty, but by its effects on the public sphere and on individual rights. This remains a rapidly developing landscape as Spain continues to refine its approach alongside evolving European standards.

== &lt;big&gt;The Right to One’s Own Image in Spain&lt;/big&gt; ==

===== The Legal Doctrine of the Right to One’s Own Image =====
Spanish law protects the “right to one’s own image” as a distinct legal interest that governs how a person’s identity—through image, voice, or other identifying features—may be used by others, particularly in media and commercial contexts. This right is codified in Organic Law 1/1982, which provides civil remedies against “illegitimate interference.”&lt;ref name=&quot;:50&quot;&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1982-11196|title=BOE-A-1982-11196 Ley Orgánica 1/1982, de 5 de mayo, de protección civil del derecho al honor, a la intimidad personal y familiar y a la propia imagen.|website=www.boe.es|access-date=2026-04-20}}&lt;/ref&gt; Article 7.5 treats as unlawful the capture or publication of a person’s image in private contexts, while Article 7.6 prohibits the use of a person’s name, voice, or image for advertising, commercial, or analogous purposes.&lt;ref name=&quot;:56&quot;&gt;{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1982-11196|title=BOE-A-1982-11196 Ley Orgánica 1/1982, de 5 de mayo, de protección civil del derecho al honor, a la intimidad personal y familiar y a la propia imagen.  arts. 7.5–7.6.|website=www.boe.es|access-date=2026-04-30}}&lt;/ref&gt;

Unlike copyright law, which protects creative works, the right to one’s own image protects the individual as the subject of representation. Even where a photograph or video is lawfully owned by a third party, the person depicted retains control over how their likeness is used.&lt;ref name=&quot;:57&quot;&gt;{{Cite journal|last=Barrnett|first=Stephen R.|date=1999|title=The Right to One's Own Image': Publicity and Privacy Rights in the United States and Spain|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=224628|journal=Am. J. Comp. L.|volume=47|pages=555}}&lt;/ref&gt; This distinction is especially important in communication law, where images circulate through media, advertising, and digital platforms independently of the underlying work.

Although commonly framed as a visual right, Spanish law explicitly extends protection beyond appearance to other identifying attributes. Article 7.6 includes the use of a person’s “name, voice, or image,” reflecting a broader concern with recognizable identity rather than strictly visual likeness.&lt;ref name=&quot;:56&quot; /&gt;

While the right originates in Article 18.1 of the Spanish Constitution, which guarantees protection of honor, privacy, and one’s own image, Spanish courts have developed it as an autonomous legal doctrine. Liability does not depend on falsity, reputational harm, or physical intrusion, but instead on the unauthorized use of identifiable personal attributes.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/pdf/1978/BOE-A-1978-40001-consolidado.pdf|title=Constitución Española art. 18.1 (1978) (Spain).}}&lt;/ref&gt; In STS 60/1998, the Supreme Court clarified that the key inquiry is whether a person can be recognized, even if not with perfect clarity.&lt;ref&gt;{{Cite web|url=https://vlex.es/vid/proteccion-fundamentales-imagen-as-17745878|title=Tribunal Supremo [T.S.] [Supreme Court], Sala Primera, Jan. 30, 1998 (RJ 1998/358) (Spain).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt; This means that partial or stylized representations, such as silhouettes, blurred images, or distinctive features, may still trigger protection if identification is possible.

At the same time, the right operates within a structured set of limits. Article 8 of Organic Law 1/1982 provides exceptions where competing interests prevail, including where there is a “predominant and relevant” cultural or informational interest or where images of public figures are captured in public settings.&lt;ref name=&quot;:50&quot; /&gt; Rather than applying rigid categories, courts evaluate whether a particular use is justified in light of its contribution to public discourse or cultural expression.&lt;ref&gt;{{Cite web|url=https://vlex.es/vid/despido-improcedente-despidos-103-3-17760168|title=Tribunal Supremo [T.S.], Sala Primera, Oct. 7, 1996 (RJ 1996/7058) (Spain).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt;

===== From Control to Commerce: How Image Rights Are Monetized =====
These principles become especially concrete in contexts like sports, where image and identity are inseparable from commercial value. Athletes’ likenesses are routinely used by clubs, sponsors, and media, but Spanish law maintains that control over that use remains with the individual.&lt;ref name=&quot;:52&quot;&gt;{{Cite web|url=https://business-school.laliga.com/en/news/image-rights-in-football|title=Image Rights in Sports {{!}} LaLiga Business School|website=Liga de Fútbol Profesional|language=en|access-date=2026-04-20}}&lt;/ref&gt; A footballer, for example, may authorize a club to use his image for team promotions, while separately licensing endorsement rights to a brand.

Spanish law structures this through a dual framework:

* Negative right: the ability to block unauthorized uses
* Positive right: the ability to license and commercially exploit one’s image

This explains why contracts in professional sports carefully define scope, duration, and purpose of any assignment.&lt;ref name=&quot;:57&quot; /&gt;

Crucially, Spanish law adopts a broad understanding of “commercial” use. This broader conception is illustrated by STS 816/1996.&lt;ref name=&quot;:53&quot;&gt;{{Cite web|url=https://vlex.es/vid/intimidad-imagen-reproduccion-autorizada-17742790|title=STS 816-1996, 7 de Octubre de 1996|website=vLex|language=es|access-date=2026-04-20}}&lt;/ref&gt; There, the City of Madrid used photographs of identifiable individuals in a public-awareness campaign promoting respect for the elderly. The Supreme Court held that the use was “publicitario” even without profit, because it relied on identifiable persons to convey its message.&lt;ref name=&quot;:53&quot; /&gt; The Court rejected the defense under Article 8.1, emphasizing that the campaign did not require the use of specific individuals’ images to achieve its purpose.

By contrast, in STS 21 December 1994, the Court allowed the reuse of a performer’s image to promote a revival of a traditional musical production.&lt;ref&gt;{{Cite web|url=https://vlex.es/vid/202673271|title=Tribunal Supremo [T.S.], Sala Primera, Dec. 21, 1994 (RJ 1994/9775) (Spain).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt; The distinguishing factor was context: the image directly related to the cultural work being promoted, and its use contributed to preserving a recognized artistic tradition. Together, these cases show that Spanish courts focus less on formal categories and more on whether the use is necessary and proportionate to its asserted purpose.

Spanish law also extends this protection to voice. In the Tom Waits case (Juzgado de Primera Instancia de Barcelona, 2006), an advertising agency hired a performer to imitate Waits’s distinctive voice after he refused to participate in a commercial. The court held this unlawful, reasoning that imitating a recognizable voice for commercial purposes exploits a person’s identity and misleads the public into believing the individual endorsed the product.&lt;ref&gt;{{Cite journal|last=PONTE|first=LUCILLE M.|date=Winter 2009|title=PRESERVING CREATIVITY FROM ENDLESS DIGITAL
EXPLOITATION: HAS THE TIME COME FOR THE NEW
CONCEPT OF COPYRIGHT DILUTION?|url=https://www.bu.edu/law/journals-archive/scitech/volume151/documents/ponte_web.pdf|journal=B.U. J. SCI. &amp; TECH. L|volume=Vol. 15.1}}&lt;/ref&gt;

===== From Exposure to Use: When the Public Can Reproduce an Image =====
A central question in this doctrine is how far public visibility allows others to reproduce a person’s image. Spanish law recognizes a strong interest in freedom of information, particularly where images contribute to reporting on matters of public concern.

In STS 28 December 1996, a newspaper published a photograph of a criminal defendant leaving court. The Supreme Court held the publication lawful because it related to a matter of public interest and contributed to informing the public about judicial proceedings.&lt;ref&gt;{{Cite web|url=https://vlex.es/vid/202746407|title=Tribunal Supremo [T.S.], Sala Primera, Dec. 28, 1996 (RJ 1996/9510) (Spain).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt; The fact that the image was taken in a public setting reinforced this conclusion.

However, public exposure does not eliminate the need for consent. This principle becomes especially important in the digital context. In STS 91/2017, a newspaper used a photograph taken from a victim’s Facebook profile when reporting a violent incident.&lt;ref name=&quot;:58&quot;&gt;{{Cite web|url=https://vlex.es/vid/667177509|title=Tribunal Supremo [T.S.], Sala Primera, Feb. 15, 2017 (RJ 2017/91) (Spain).|website=vLex|language=es|access-date=2026-04-30}}&lt;/ref&gt; The Court held that this violated the right to one’s image, emphasizing that making a photograph accessible online does not amount to consent for its reuse. Consent must be specific to each use and cannot be inferred from general availability.&lt;ref name=&quot;:58&quot; /&gt;

The limits of permissible use become even clearer in cases involving dignity and suffering. In STC 231/1988, the Constitutional Court held that distributing footage of a bullfighter dying after being gored violated the privacy rights of his widow.&lt;ref&gt;{{Cite web|url=https://www.boe.es/buscar/doc.php?id=BOE-T-1988-29203#:~:text=2.%20Do%C3%B1a%20Isabel%20Pantoja%20Mart%C3%ADn,%20ahora%20recurrente,derecho%20a%20la%20intimidad%20y%20a%20la|title=BOE-T-1988-29203 Sala Segunda. Sentencia 231/1988, de 2 de diciembre. Recurso de amparo 1.247/1986. Contra Sentencia de la Sala Primera del Tribunal Supremo que anula la dictada en apelación por la Audiencia Territorial de Madrid, en autos sobre vulneración del derecho a la intimidad. Voto particular.|website=www.boe.es|access-date=2026-04-30}}&lt;/ref&gt; Although the event occurred in a public arena, the Court concluded that the dissemination of images capturing extreme distress crossed the boundary of acceptable informational use.

As of now, the Spanish right to one’s own image is best understood as a doctrine of controlled visibility. It protects an individual’s authority over how they are represented, even in public-facing contexts such as media, sports, and digital platforms. While the law accommodates competing interests, such as news reporting, cultural expression, and satire, it consistently resists the idea that visibility alone permits unrestricted use.

In an environment where images circulate rapidly and widely, this framework ensures that identity remains anchored in the individual rather than absorbed into the commercial or informational systems that reproduce it.

== References ==
[[Category:Communication in Europe|Law in Spain]]
[[Category:Law in Europe]]
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      <comment>/* See also */ ;Wikimedia Commons 
* [https://diff.wikimedia.org/2026/05/04/eduwiki-workshop-highlights-practical-uses-of-wikimedia-commons-in-education/ EduWiki workshop highlights practical uses of Wikimedia Commons in education ]</comment>
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&lt;div style=&quot;text-align: center&quot;&gt;

[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]&lt;br&gt;
Friday 24 April, 2026 11:00 - 12:00 AEST

[https://utsmeet.zoom.us/j/84179400467 Live] (Zoom)

[https://docs.google.com/presentation/d/1aNPMHQgYoKaDuOqLfjUa44vb0kySmfYS9glWop0zOTM/edit?usp=sharing Slides] (Google)

&lt;!-- Slides TBA (Google)&lt;br&gt;
Video TBA (YouTube; 53:37 mins including Q&amp;A)
[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post]
[https://twitter.com/jtneill/status/1768516693553565884 (example) X post]
--&gt;
&lt;/div&gt;
{{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}}
==Overview==
Many student assignments are written for one person, read once, and then never read again. 

In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. 

Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio.

The session will explore: 
* What open wiki assignments look like in practice, and where they go wrong
* The realities of working in publicly editable spaces (including having work changed or deleted)
* Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around

This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching.

==Tasks==
===Task 1: Questions to consider===
# What is your experience with wikis?
# Have you contributed to open wikis such as Wikipedia?
# Would you like to share open educational educational resources?
# Could editing a wiki provide useful learning for your students?
# What are the opportunities for open wikis in higher education?

===Task 2: Explore Wikiversity===
# Visit the [[Main page|English Wikiversity]] homepage
# Read [[Wikiversity:Welcome|Welcome to Wikiversity]]
# Visit random [[Special:RandomPage|random Wikiversity page]]s
# Explore the [[Motivation and emotion/Book|motivation and emotion book project]]

===Task 3: Set up and log in to a Wikimedia Foundation account===
# [[Special:CreateAccount|Create a Wikimedia Foundation account]] (if you don't already have one) 
# [[Special:UserLogin|Login]] once you have an account

===Task 4: Edit your Wikiversity user page===
# [[Special:MyPage|Edit your Wikiversity user page to introduce yourself]]

==I love wikis==
[[File:I love wiki.svg|right|150px|thumb|'''Figure 1'''. I love wikis because they are the simplest public webpages which are editable.]]
I love wikis, especially open wikis, especially the Wikimedia Foundation sister projects, especially Wikiversity, and especially the English language Wikiversity.

I love wikis because they are the simplest public webpages that are editable by anyone. I love wikis because they allow crowd-sourced, consensus-based knowledge development.

I use wikis for almost everything including to share presentations, research, teaching materials including lectures, tutorials, and workshops, and as a platform for student-authored assignments.

==But ... what is the problem?==
Disposable assignments set all students the same task to show knowledge/competence or learning (see Figure 2). Disposable assignments are a problem because they:
* are over-used
* provide limited empowerment
* waste human productivity

Submitted assignments are generally only used for marking and feedback purposes and never see the light of day.

Disposable assignments are missed opportunities. Student work could be shared with the world by publishing to the open web and being added to an eportfolio showcasing a student's capacities. However, sharing multiple versions of the same disposable assignment is of limited value and risks academic integrity issues.
[[File:Disposable versus reusable assignments.svg|center|600px|'''Figure 2'''. Disposable versus reusable assignments]]

==Reusable assignments==
Enter reusable assignments. What if an assignment involved each student tackling a unique task? This is a reusable assessment because it:
* continually creates new tasks and 
* students produces valuable resources:
** contribute valuable disciplinary resources to the knowledge commons
** serve as open artifacts of student learning

At first glance reusable assessments may seem onerous on the teacher (to create novel tasks). But there can be many variations on a central theme or pattern, such as [[Motivation and emotion/Book|1000s of student-authored online book chapters about the science of motivation and emotion]] can be applied to improving people's everyday lives.

==Why use wikis?==
[[w:Wiki|Wiki]]s were developed in the early days of the world wide web (1994; see [[w:History of wikis|history of wikis]]) as the simplest web page that anyone can edit (see Figure 1). Today, wiki technology serves as the foundation for the global open knowledge projects developed by the [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF), the best known of which is [[w:|Wikipedia]]. Along with a dozen or so [[Wikiversity:Sister projects|sister projects]], these wikis have in common the lofty goal of making the sum of all human knowledge freely available to all—and making that knowledge editable by anyone. WMF project pages are oft-visited, well ranked by search engines, updatable, used to train artificial intelligence, and their open licensing allows the material to be re-used for other purposes.

==Open wikis and higher education==
[[File:Wiki project case study onion diagram.svg|right|195px|thumb|'''Figure 2'''. An onion layer model of open wiki assignments for authentic learning]]

Wiki technology enables grand social experiments. Like universities, wikis are great places to collaborate and engage in learning and research activities (see Figure 2). Wiki projects can be used by teachers and students to curate disciplinary knowledge and develop collaboration skills by engaging in collaborative editing and commenting on each other’s work.

Wiki content is immediately available on the web and can be edited by anyone. In this way the open educational resources are iteratively and rapidly improved. The radical transparency of open editing can initially seem daunting for staff and students schooled in an all rights reserved normative culture, but an open approach quickly empowers participants' capacity and confidence in their capacity to engage in and contribute directly to the knowledge commons and have agency in sharing and improving this work.

==Intellectual property and copyright==
Students own the copyright to their work. Staff should check their instituational policies, but there is increasing recognition of, and support for, open access publishing of teaching and research.

WMF project content is openly licensed and free to use ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) to allow maximum re-usability and meeting the UNESCO definition of free cultural works. Therefore, students contributing to open wikis must willing agree to license their work in this way, otherwise alternative assignment format options should be available.

==Wiki assignments==
Wiki-based assignment formats vary depending on:
* Subject area (e.g., humanities, sciences, professional fields)
* Level of study (e.g., informal, K-12, undergraduate, postgraduate, professional learning)
* Intended learning outcomes and development of generic skills (e.g., critical thinking, communication, digital literacy)

Typical tasks involve creating, curating, or improving open educational resources such as:
* Books (e.g., open textbooks)&lt;!-- — Wikibooks --&gt;
* Data (e.g., linked data items and data analysis)&lt;!-- —  — Wikidata --&gt;
* Conferences (e.g., home page, applications, abstracts, program, presentations)&lt;!-- —  — Wikiversity --&gt;
* Encyclopedic articles&lt;!-- —  — Wikipedia --&gt;
* Eportfolios&lt;!-- —  --&gt;
* Essays and analytical reports&lt;!-- —  — Wikiversity --&gt;
* Fact sheets and study guides&lt;!-- —  — Wikiversity --&gt;
* Images, audio, and video (with metadata and licensing)&lt;!-- —  — Wikimedia Commons --&gt;
* Journal-style articles and literature reviews&lt;!-- —  — Wikiversity --&gt;
* Learning activities (e.g., lectures, tutorials, workshops)&lt;!-- —  — Wikiversity --&gt;
* Manuals, tutorials, and how-to guides&lt;!-- —  — Wikiversity --&gt;

===Open wikis and learning management systems===
Open educational wikis can serve as [[w:Content management system|content management systems]] for hosting teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. Wikis can support the development of open textbooks. Compared to popular institutionally supported textbook platforms for open textbooks such as [[w:PressBooks|PressBooks]], there are no fees to use Wikimedia projects and they enable more diverse, collaborative, and participatory knowledge production.&lt;ref group=&quot;note&quot;&gt;In the context of [[w:Tertiary education in Australia|Australian higher education]], platforms such as PressBooks are typically staff-controlled, with limited opportunities for student authorship and co-creation.&lt;/ref&gt;

===Wikis as collaborative knowledge systems===
Wikis foster a pragmatic, solution-focused technology platform and culture for collaborative knowledge development. Wikis support long-term knowledge preservation. For students, an open wiki assignment ensures that their work remains publicly available and that they retain access to course materials  materials beyond graduation. For staff, access to open wiki materials continues regardless of institutional affiliation.

====Always open for improvement====
[[File:Linus' law.png|thumb|right|400px|'''Figure 4'''. Linus' law that &quot;given enough eyeballs, all bugs are shallow&quot; is applicable to open wiki student assignment projects. By having student work available to others during its development, peers can contribute and provide feedback, leading to a better quality product.]]

Wiki content is iteratively improved by having many eyeballs, brains, and fingers to click edit, change, and publish. Linus Torvald was talking about code when he said that &quot;given enough eyeballs, all bugs are shallow&quot; (see Figure 4), but it also applies to open wiki content which invites scrutiny and encourages improvements made by anyone.

====Version control and editing history====
A notable feature of wikis is that every page has a complete, searchable edit history. Each revision can be reviewed and, if necessary, reverted, ensuring that no content is permanently lost. Most edits incrementally improve the quality of a page; however, a small proportion are unhelpful and are therefore undone. As a rough guide, approximately 95% of edits are retained, while around 5% are reverted or deleted.

====Handling disagreement====
On wikis, disagreements about content are addressed through open discussion and consensus-building. This creates a distinctive collaborative environment in which students develop core skills in argumentation, communication, and negotiation.

====Forking====
Wiki content can also be readily forked, similar to open-source software, enabling alternative versions to evolve in parallel.

====Languages====
There is also a need for translation and development of open knowledge materials in different languages.

==Wikimedia projects==
This section describes Wikimedia student assignments, including their affordances and what to be wary about. It then explains how assignments can be conducted on Wikipedia and WMF sister projects. 

===Wikipedia===
[[File:Wikipedia-logo-v2.svg|right|150px|thumb|Wikipedia logo]]
The most successful and notable open knowledge educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists (see [[w:Wikipedia:Student assignments|Wikipedia:Student assignments]].

The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. 

However, I would cast the net wider than Wikipedia because:
* Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence.
* Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources.

For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects.

===What does a collaborative open wiki project with students involve?===
An open wiki higher education student assignment generally involves:
* Students contributing discipline-relevant content to the global knowledge commons via a Wikimedia Foundation sister project 
* Assignment tasks centre on producing and refining knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences
* Outputs can include text, media, data, and learning resources  
* Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages])  

===Beyond Wikipedia: WMF sister projects===
Opportunities for students to contribute open knowledge extend beyond encyclopedic work on Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]] (see Table 1). These platforms provide public environments for producing, curating, and sharing openly licensed scholarly work. These less well known projects offer targetted, specific environments for specific discipline foci and learning objectives. Wikiversity is notable because it serves as the default or main project for educational work because its [[Wikiversity:Mission|mission]] is closely aligned with the purpose of [[w:higher education|higher education]].

'''Table 1. Wikimedia Foundation Sister Projects'''

{{Sisterprojects/Projects}}
Table 2 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development.

Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources.

{| class=&quot;wikitable&quot; style=&quot;margin: 0 auto;&quot;
|+ Table 2. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments
! Project
! Purpose
! Example assignments
|-
| [[b:|Wikibooks]]
| New books (e.g., textbooks)
| 
* Contribute to development of an open textbook
* Curate and improve existing OER book chapters
* Package a series of related articles into a new book
|-
| [[commons:|Wikimedia Commons]]
| Images, audio, and video
|
* Contribute high-quality educational media
* Improve metadata and categorisation
* Create educational diagrams and illustrations
* Upload field recordings or interviews
|-
| [[d:Wikidata|Wikidata]]
| Structured, linked open data
|
* Create and curate datasets
* Link concepts across Wikimedia projects
* Model relationships between entities
* Support data-driven research and analysis
|-
| [[q:|Wikiquote]]
| Quotations
|
* Curate and improve text quotes from primary sources such as political speeches
* Create categories for quotes by theme or topic
* Add citations and verification to existing quotes
|-
| [[species:|Wikispecies]]
| Taxonomy and species classification
|
* Curate and improve taxonomic entries for species
* Add citations for classification and nomenclature
* Contribute information about newly described species
* Improve links between species and related Wikimedia projects
|-
| [[s:Main Page|Wikisource]]
| Primary texts and historical documents
|
* Transcribe and proofread source texts
* Annotate and contextualise historical documents
* Curate thematic collections of primary sources
|-
| [[v:|Wikiversity]]
| Learning, teaching, and research
|
* Create open educational resources
* Develop teaching materials (e.g., lesson plans, self-assessment quizzes)
* Publish student research project summaries
* Improve existing learning resources by adding new text and multimedia
|-
| [[voy:Main Page|Wikivoyage]]
| Travel guides and geographic knowledge
|
* Develop place-based guides (e.g., regions, cities)
* Contribute cultural, historical, or environmental information
* Integrate fieldwork or experiential learning outputs
|-
| [[wikt:Main Page|Wiktionary]]
| Lexical and linguistic resources
|
* Create and refine dictionary entries
* Analyse word meanings, usage, and etymology
* Contribute multilingual translations and examples
|-
| [[w:|Wikipedia]]
| Encyclopedic information
|
* Contribute to articles related to the class topic where a gap exists
* Improve the quality and accuracy of existing articles
* Add citations and references to unverified text
* Curate and improve a category of articles related to a specific subject area
|}

==Open wiki assignments==
Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available.

==FAQ about open wiki assignments==
Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include:
* '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s).
* '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators.
* '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions.
* '''Editing wikis is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]].
*'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. 
* '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research.

==Advantages of open wiki assignments==
Advantages of open wiki assignments include:
* '''Perpetuity''' - ongoing availability of resources
* '''Linkability''' - cross-linking of projects and external resources
* '''Editability''' - resources can be improved by anyone
* '''Discussability''' - each resource has a discussion page
* '''Showability''' - resources showcase curator skills and knowledge
* '''Transparency''' - resource edit history and can be reviewed
* '''Forkability''' - open licence allows development of alternative resources

==Examples==
Here are some examples of open wiki assignments:
* [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia 
* [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia
* [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA

Whilst not assignments per se, these innovative open wiki resources may inspire:
* [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries

==Activities==
* Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]]
* Edit your [[Help:User page|Wikiversity user page]]
* Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]]
* Brainstorm what you or your students could contribute
* Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support

==Bio==
[[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' [[User:Jtneill/Teaching/Philosophy|teaching philosophy]].

==See also==
;Wikimedia Foundation
* [[meta:Education|Education]] (Global WMF hub)
* [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia)

;Wikimedia Commons 
* [https://diff.wikimedia.org/2026/05/04/eduwiki-workshop-highlights-practical-uses-of-wikimedia-commons-in-education/ EduWiki workshop highlights practical uses of Wikimedia Commons in education ]


;Wikipedia
* [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta)
* [[w:Wikipedia:Student assignments|Wikipedia:Student assignments]] (Wikipedia)
* [https://wikiedu.org/ Wiki Education] (a separate non-profit that supports North American faculty and teachers to conduct student assignment projects on Wikipedia)

;Wikiversity
* [[Motivation and emotion/Tutorials/Wiki editing|A Wikiversity editing tutorial]] (Motivation and emotion)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Neill, 2024; article)
* [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page)
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation)

[[Category:User:Jtneill/Presentations/Open education]]
[[Category:User:Jtneill/Presentations/Wikiversity]]

== Notes ==
&lt;references group=&quot;note&quot; /&gt;&lt;!--
== References ==
&lt;references /&gt;
--&gt;</text>
      <sha1>9tgnvsavk3r86yg85cfld2nrezlfo76</sha1>
    </revision>
  </page>
  <page>
    <title>Intuitive Calculus</title>
    <ns>0</ns>
    <id>329025</id>
    <revision>
      <id>2810638</id>
      <parentid>2809822</parentid>
      <timestamp>2026-05-20T18:26:31Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>may be doing more notes today</comment>
      <origin>2810638</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8724" sha1="almsxxoyaqfraxgv8tfxk8osd802mmd" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>almsxxoyaqfraxgv8tfxk8osd802mmd</sha1>
    </revision>
    <revision>
      <id>2810654</id>
      <parentid>2810638</parentid>
      <timestamp>2026-05-20T20:45:57Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* 5/20/2026 */ all I could get to for right now</comment>
      <origin>2810654</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9033" sha1="78ei1gcd4wsebbf34h2eqb2jelo2at9" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>78ei1gcd4wsebbf34h2eqb2jelo2at9</sha1>
    </revision>
    <revision>
      <id>2810703</id>
      <parentid>2810654</parentid>
      <timestamp>2026-05-21T01:24:50Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* 5/20/2026 */ https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf</comment>
      <origin>2810703</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9288" sha1="1lpadtorbxkpzxcob3tbn5x0swrapws" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>1lpadtorbxkpzxcob3tbn5x0swrapws</sha1>
    </revision>
    <revision>
      <id>2810706</id>
      <parentid>2810703</parentid>
      <timestamp>2026-05-21T01:42:55Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* 5/20/2026 */</comment>
      <origin>2810706</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10138" sha1="npehw9x8u54l0ognh5q84m5npjjexm1" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* '''Fermet's Theorem =''' If a real-valued function, f(x), is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval (a, b) and f(x) has a maximum OR minimum at c ∈ (a, b), then f'(c) = 0&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially belongs to/inside/a member of. For example, c ∈ (a, b) → &quot;the number c is inside the interval between a and b&quot;.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>npehw9x8u54l0ognh5q84m5npjjexm1</sha1>
    </revision>
    <revision>
      <id>2810712</id>
      <parentid>2810706</parentid>
      <timestamp>2026-05-21T02:01:54Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* 5/20/2026 */</comment>
      <origin>2810712</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10194" sha1="3s2bfexak067r1i792hfxy1orpa6gfo" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, f(x), is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval (a, b) and f(x) has a maximum OR minimum at c ∈ (a, b), then f'(c) = 0&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially belongs to/inside/a member of. For example, c ∈ (a, b) → &quot;the number c is inside the interval between a and b&quot;.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>3s2bfexak067r1i792hfxy1orpa6gfo</sha1>
    </revision>
    <revision>
      <id>2810775</id>
      <parentid>2810712</parentid>
      <timestamp>2026-05-21T11:40:53Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* 5/20/2026 */</comment>
      <origin>2810775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10196" sha1="klq1j2ggt991fkh7wzm8vxjywdx04tx" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, f(x), is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval (a, b) and f(x) has a maximum OR minimum at c ∈ (a, b), then f'(c) = 0&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially &quot;belongs to/inside/a member of.&quot; For example, c ∈ (a, b) → &quot;the number c is inside the interval between a and b&quot;.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>klq1j2ggt991fkh7wzm8vxjywdx04tx</sha1>
    </revision>
    <revision>
      <id>2810776</id>
      <parentid>2810775</parentid>
      <timestamp>2026-05-21T11:42:06Z</timestamp>
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        <username>Atcovi</username>
        <id>276019</id>
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      <comment>/* 5/20/2026 */</comment>
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      <text bytes="10386" sha1="g73355l2y3svg09kkay646xe9uoyltx" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|pages=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
{{Notice|1=
'''5/14/2026''' - STOPPING POINT&lt;br&gt;
To watch for later: https://www.youtube.com/watch?v=AOKoo_nQSts (6:01)

To read for later: https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?params=/context/triumphs_calculus/article/1011/&amp;path_info=M05_Fermats_Method_for_Finding_Maxima_and_Minima_2022_05_17.pdf&amp;cs=1&amp;hl=en-US&amp;biw=1280&amp;bih=631.3333740234375}}
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maxima occurs). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, &lt;math&gt;f(x)&lt;/math&gt;, is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval &lt;math&gt;(a, b)&lt;/math&gt; and &lt;math&gt;f(x)&lt;/math&gt; has a maximum OR minimum at &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;f'(c)&lt;/math&gt; = &lt;math display=&quot;inline&quot;&gt;0&lt;/math&gt;&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially &quot;belongs to/inside/a member of.&quot; For example, &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt; → &quot;the number c&lt;math&gt;&lt;/math&gt; is inside the interval between &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt;&quot;.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>g73355l2y3svg09kkay646xe9uoyltx</sha1>
    </revision>
  </page>
  <page>
    <title>User:Atcovi/OGM &amp; Suicide/The Paper</title>
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      <id>2810616</id>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Introduction */ spelling</comment>
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      <text bytes="17719" sha1="qjrgt83tzmnkqqh3y812mkk8i5bfv9c" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model protrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to have a significant contributing factor in worsening suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness),and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be a significant contributor in high-risk/depressed populations. Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17975" sha1="7pzmwff9b4mjlg1nogdbr3nbldxt1lt" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model protrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to have a significant contributing factor in worsening suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
'''Meat of the conclusion''': OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness), and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). 

'''Limitations''': Despite significant research indicating OGM's unique contribution to suicidal ideation, OGM appears to be limited as a significant contributor for high-risk/depressed populations. Other limitations include that the studies collected were mostly cross-sectional and the causal direction between OGM and suicidal ideation is still somewhat unclear.

'''Future direction''': Future research should look into OGM's place in suicidal models and assess whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial across diverse populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="18283" sha1="fkq35r4gxzbakr7cwyvm7pbk6peoxhg" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model protrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to have a significant contributing factor in worsening suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
{{Notice|
* reinforce your thesis
* clarify implications
* acknowledge limitations
* propose future directions}}
'''Meat of the conclusion''': OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness), and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). From the literature, OGM appears to be a significant contributor to suicidal ideation for high-risk/depressed populations. This paper illustrates where OGM can contribute to suicidal ideation according to the IMV model to better inform researchers on identifying relevant risk factors.  

'''Limitations''': Limitations include the cross-sectional research method of the studies, the causal direction between OGM and suicidal ideation is still somewhat unclear, and the conclusion derived from this research cannot be generalized beyond high-risk populations.

'''Future direction''': Future research should look into empirical testing of OGM and entrapment and to test if OGM moderates the transition from entrapment to suicidal ideation. Future research could also look into whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial for high-risk populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="18170" sha1="h24wnrvsxq5a6dmz6j8c9tn2787jpnz" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model protrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to have a significant contributing factor in worsening suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
'''Meat of the conclusion''': OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. This can lead to negative Motivational Moderators (such as thwarted belongingness and perceived burdensomeness), and the transition from suicidal ideation to suicidal behavior within the Volitional Phase (where suicidal behavior is present). From the literature, OGM appears to be a significant contributor to suicidal ideation for high-risk/depressed populations. This paper illustrates where OGM can contribute to suicidal ideation according to the IMV model to better inform researchers on identifying relevant risk factors.  

'''Limitations''': Limitations include the cross-sectional research method of the studies, the causal direction between OGM and suicidal ideation is still somewhat unclear, and the conclusion derived from this research cannot be generalized beyond high-risk populations.

'''Future direction''': Future research should look into empirical testing of OGM and entrapment and to test if OGM moderates the transition from entrapment to suicidal ideation. Future research could also look into whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial for high-risk populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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      <text bytes="17930" sha1="ov5bksrjmpszfwbymgvut370fj4704i" xml:space="preserve">'''Major objective''': OGM may function as a context-dependent cognitive vulnerability that contributes to suicidal escalation through mechanisms embedded within the IMV model, particularly in high-risk populations. Through mechanisms outlined in the IMV framework, OGM may contribute to increased suicidal ideation within high-risk populations. 

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing the '''Integrated Motivational-Volitional (IMV) model''' provides a theoretical cognitive framework to argue that OGM intensifies suicidal escalation through moderators described in the model. 

The IMV model protrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM contributes to suicidal ideation, OGM appears to impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation within the IMV framework in high-risk population. 

==Mechanisms of OGM==
The inability to retrieve specific memories from one's autobiographical memory may lead to inefficient problem-solving abilities, which can impact one's ability to deal with difficult situations as they lack past experiences to rely on&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Arie|first=Miri|last2=Apter|first2=Alan|last3=Orbach|first3=Israel|last4=Yefet|first4=Yael|last5=Zalzman|first5=Gil|date=2008-01-01|title=Autobiographical memory, interpersonal problem solving, and suicidal behavior in adolescent inpatients|url=https://www.sciencedirect.com/science/article/pii/S0010440X07000922|journal=Comprehensive Psychiatry|volume=49|issue=1|pages=22–29|doi=10.1016/j.comppsych.2007.07.004|issn=0010-440X}}&lt;/ref&gt;. Failing to deal with difficult situations can drive an individual to hopelessness. In 1986, an article by Mark J. Williams and Keith Broadbent stipulated that individuals who recently attempted suicide had biased latencies in autobiographical memory retrieval and had reduced specificity in responses to especially positive cues&lt;ref&gt;{{Cite web|url=https://www.tandfonline.com/action/cookieAbsent|website=www.tandfonline.com|doi=10.1080/02699938808410925|access-date=2026-05-08}}&lt;/ref&gt;. OGM may not have a direct effect on suicide, but the evidence suggests that OGM intensifies the risk of suicidal ideation through deteriorating cognitive functioning.   

Rumination involves maladaptive dwelling on one's past negative emotions and feelings&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Treynor|first=Wendy|last2=Gonzalez|first2=Richard|last3=Nolen-Hoeksema|first3=Susan|date=2003-06-01|title=Rumination Reconsidered: A Psychometric Analysis|url=https://doi.org/10.1023/A:1023910315561|journal=Cognitive Therapy and Research|language=en|volume=27|issue=3|pages=247–259|doi=10.1023/A:1023910315561|issn=1573-2819}}&lt;/ref&gt;, and is associated with suicidal behavior/ideation&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational–volitional model of suicidal behaviour|url=https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0268|journal=Philosophical Transactions of the Royal Society B: Biological Sciences|language=en|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=0962-8436|pmc=6053985|pmid=30012735}}&lt;/ref&gt;. One may reflect on their negative emotions and question such emotions in an abstract manner (&quot;How did I get to feel this way?&quot;&lt;ref name=&quot;:0&quot; /&gt;, &quot;Why did this happen to me?&quot;&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|date=2012-02|title=The mechanisms underlying overgeneral autobiographical memory: an evaluative review of evidence for the CaR-FA-X model|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3246105/|journal=Clinical Psychology Review|volume=32|issue=1|pages=34–48|doi=10.1016/j.cpr.2011.10.003|issn=1873-7811|pmc=3246105|pmid=22142837}}&lt;/ref&gt;), which causes one's memory retrieval to capture negative intermediate conceptual information (ex, &quot;I'm a failure&quot;) instead of specific memories. Repeated rumination strengthens these negative self-beliefs, leading to frequent capture of negative conceptual themes and impeding memory retrieval. This association of rumination and general memory aligns with the CaR-FA-X model&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;.  

Within the IMV model, I propose that OGM may contribute to entrapment (a perceived sense of being trapped by defeat/humilitation) through impaired autobiographical memory, repeated rumination, and hopelessness. Positive factors, such as motivation to live, positive future thinking, and belongingess can offset the transition of entrapment → suicidal ideation, though negative factors from '''Motivational Moderators''' (MM), such as thwarted belongingness, very little social support, and perceived burdensomeness, may increase the chance of entrapment converting into suicidal ideation&lt;ref name=&quot;:1&quot; /&gt;.  

==OGM as a Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref name=&quot;:2&quot; /&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref name=&quot;:2&quot; /&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations. OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref name=&quot;:2&quot; /&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. Across the scientific literature, OGM is suggested to have a significant contributing factor in worsening suicidal ideation, but appears to be have a more prevalent predictive relevance in higher-risk populations. 

=== OGM → Suicidal ideation ===
The research suggests that OGM may not just be associated with depression, but is may also be a contributing factor towards suicidal ideation. Jiang et. al (2020) found in a study of 365 participants, with roughly 51% of the participants clinically depressed while the other roughly 49% of participants were classified as &quot;healthy&quot;, that OGM had an increased presence in the depressed group vs. the &quot;healthy&quot; group. WSI (worst suicidal thoughts one has ever had [at a certain point]) and CSI (current point of suicidal ideation) were significantly affected by OGM in the depressed group. OGM was also found to be a mediator between CSI and childhood trauma in depressed patients. As OGM leads to negative memory biases and, therefore, the maintenance of a negative mental state, the researchers suggested that OGM may be a consistent contributor to suicidal ideation in depressed patients&lt;ref name=&quot;:3&quot; /&gt;. These findings are corroborated by a more recent (2025) study on depressed patients with varying levels of SI, where the researchers concluded that OGM may be &quot;a maladaptive cognitive avoidance strategy&quot; rather than simply a deterioration in memory. Zhu et. al (2025) further explain that individuals with OGM have a difficult time recalling positive memories, which reinforce negative recollections, spurring hopelessness and the transition to suicidal ideation&lt;ref&gt;{{Cite journal|last=Zhu|first=Ying|last2=Yin|first2=Qianlan|last3=Xu|first3=Huijing|last4=Xiao|first4=Fang|last5=Jiang|first5=Qian|last6=Liang|first6=Meng|last7=Cheng|first7=Qi|last8=Liu|first8=Taosheng|date=2025-11-24|title=Speech feature identification model for depressed individuals with suicidal ideation based on autobiographical memory|url=https://doi.org/10.1186/s12888-025-07635-0|journal=BMC Psychiatry|language=en|volume=25|issue=1|pages=1154|doi=10.1186/s12888-025-07635-0|issn=1471-244X|pmc=12713288|pmid=41286790}}&lt;/ref&gt;. 

In a 2008 study on autobiographical memory, interpersonal problem-solving skills, and suicidal behaviour in adolescents and young adults, Arie et. al (2008) found that OGM was significantly associated with hopelessness and poor problem-solving abilities in adolescents. This suggests that being able to retrieve specific memories in one's autobiographical memory improves problem-solving skills, as they are able to draw back from past experiences to address challenging interpersonal situations&lt;ref name=&quot;:4&quot; /&gt;. Kaviani et. al (2011) found that depressed individuals with more severe suicidal ideation levels had more difficulty in retrieving specific thoughts in comparison to depressed individuals with less severe suicidal ideation&lt;ref&gt;{{Cite journal|last=Kaviani|first=H.|last2=Rahimi|first2=M.|last3=Rahimi-Darabad|first3=P.|last4=Naghavi|first4=K. Kamyar H.|date=2003|title=HOW AUTOBIOGRAPHICAL MEMORY DEFICITS AFFECT PROBLEM-SOLVING IN DEPRESSED PATIENTS|url=https://acta.tums.ac.ir/index.php/acta/article/view/2663|journal=Acta Medica Iranica|language=en-US|pages=194–198|issn=1735-9694}}&lt;/ref&gt;. These findings suggest that OGM may play a unique factor in contributing to suicidal ideation through maladaptive cognitive processing rather than being merely a symptom of depression.

=== Emphasis on High-Risk Population ===
Despite the importance of OGM and the findings indicating its potential amplification of suicidal ideation, OGM does not appear to be a consistent detriment in low-risk populations. 

Crane et. al (2016) conducted a longitudinal study of 5792 adolescents from ages 13 to 16 and found no significant findings that OGM played a direct or interactive role with depression, suicidal ideation, and self-harm (when accounted for confounding variables) in a general population&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. The authors concluded that OGM appears to be more clinically meaningful in high-risk populations that are already cognitively vulnerable through depression and/or psychopathology. The OGM x Stress interaction theory is supported by another longitudinal study done on 174 Caucasian adolescents by Stange et. al (2012), where they found that OGM was found to be a vulnerability to adolescents with depression (especially emotional maltreatment)&lt;ref&gt;{{Cite journal|last=Stange|first=Jonathan P.|last2=Hamlat|first2=Elissa J.|last3=Hamilton|first3=Jessica L.|last4=Abramson|first4=Lyn Y.|last5=Alloy|first5=Lauren B.|date=2013-02|title=Overgeneral autobiographical memory, emotional maltreatment, and depressive symptoms in adolescence: evidence of a cognitive vulnerability-stress interaction|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3530666/|journal=Journal of Adolescence|volume=36|issue=1|pages=201–208|doi=10.1016/j.adolescence.2012.11.001|issn=1095-9254|pmc=3530666|pmid=23186994}}&lt;/ref&gt;. 

These findings, alongside with Jiang et. al (2020), suggest that OGM is context-dependent and may play a significant role in the development of depression and/or suicidal ideation if the individual is already susceptible for depression and/or mental disorders. This aligns with the diathesis-stress model, suggesting a set of factors interact with pre-existing vulnerabilities to produce a &quot;disordered state&quot;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5|pages=24-25}}&lt;/ref&gt;.

==Conclusion==
'''Meat of the conclusion''': OGM has been found to be a cognitive vulnerability for suicidal ideation amongst clinically high-risk samples. Within the IMV model, OGM appears to contribute to entrapment through impaired retrieval of past experiences, repeated rumination, and hopelessness. From the literature, OGM appears to have a meaningful contribution to suicidal ideation for high-risk/depressed populations. This paper illustrates where OGM can contribute to suicidal ideation according to the IMV model to better inform researchers on identifying relevant risk factors.  

'''Limitations''': Limitations include the cross-sectional research method of the studies, the causal direction between OGM and suicidal ideation is still somewhat unclear, and the conclusion derived from this research cannot be generalized beyond high-risk populations.

'''Future direction''': Future research should look into empirical testing of OGM and entrapment and to test if OGM moderates the transition from entrapment to suicidal ideation. Future research could also look into whether therapies targeting OGM, such as Memory Specificity Training (MEST), would be beneficial for high-risk populations in reducing the risk of suicidal ideation.

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
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    <title>Athena problem</title>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem in base ''b'' covers the Sierpiński problem in base ''b'' and the Riesel problem in base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'', since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b''):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits): {11}

Base 3: 3 primes (the largest of which has 3 digits): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 709070, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits, see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem in base ''b'' covers the Sierpiński problem in base ''b'' and the Riesel problem in base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'', since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b''):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits): {11}

Base 3: 3 primes (the largest of which has 3 digits): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 709070, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>gwb6no0lsulgzawhieixp8lwph9li81</sha1>
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  <page>
    <title>Wikiversity:Candidates for Bureaucratship/Koavf</title>
    <ns>4</ns>
    <id>329564</id>
    <revision>
      <id>2810760</id>
      <parentid>2810528</parentid>
      <timestamp>2026-05-21T10:16:17Z</timestamp>
      <contributor>
        <username>Leaderboard</username>
        <id>974929</id>
      </contributor>
      <comment>/* Voting */ +1</comment>
      <origin>2810760</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3045" sha1="bw6vrh79uwrsnzrz11u80iafkte4skd" xml:space="preserve">=== {{User|Koavf}} ===
Per the [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;oldid=2808455#Call_for_custodians_and_bureaucrats call] made by {{user|Jtneill}}, I am self-nominating for bureaucrat. I have had advanced user rights here for years and have been an bureaucrat on &lt;del&gt;[[:oversight:]]&lt;/del&gt;&lt;ins&gt;[[:outreach:]]&lt;/ins&gt; and a CheckUser on [[:species:]], among other wikis where I am an admin/sysop. I would be happy to help here as needed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:48, 12 May 2026 (UTC)
==== Questions ====
:Can you clarify what you mean with the [[oversight:]] red link? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:21, 12 May 2026 (UTC)
:: Pinging [[User:Koavf|Koavf]] to my question. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:52, 14 May 2026 (UTC)
:::Whoops. It was [[:outreach:]]. I'm also a bureaucrat on [[:s:mul:]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:54, 14 May 2026 (UTC)
:::: No problem, thank you for the clarification. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:16, 14 May 2026 (UTC)

==== Comments ====
==== Voting ====
* {{support}} Prolific contributor to Wikimedia projects, with extensive and well-respected administrative experience and proven willingness to help English Wikiversity. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:41, 12 May 2026 (UTC)
* {{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:26, 13 May 2026 (UTC)
* {{support}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:16, 14 May 2026 (UTC)
* {{Support}}, as per Jtneill. [[User:Tommy Kronkvist|Tommy Kronkvist]] ([[User talk:Tommy Kronkvist|discuss]] • [[Special:Contributions/Tommy Kronkvist|contribs]]) 19:14, 19 May 2026 (UTC)
* {{support}} no issues. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:43, 20 May 2026 (UTC)
* {{support}} though I don't contribute much here so feel free to discard if it doesn't count. [[User:Leaderboard|Leaderboard]] ([[User talk:Leaderboard|discuss]] • [[Special:Contributions/Leaderboard|contribs]]) 10:16, 21 May 2026 (UTC)
[[Category:Nominations for Bureaucratship|Koavf]]</text>
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  <page>
    <title>Wikiversity:Candidates for Bureaucratship/Atcovi</title>
    <ns>4</ns>
    <id>329572</id>
    <revision>
      <id>2810759</id>
      <parentid>2809568</parentid>
      <timestamp>2026-05-21T10:15:35Z</timestamp>
      <contributor>
        <username>Leaderboard</username>
        <id>974929</id>
      </contributor>
      <comment>/* Voting */ +1</comment>
      <origin>2810759</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4915" sha1="el3hqslkee9b79bxdn6hnpngzl4rg2j" xml:space="preserve">=== {{User|Atcovi}} ===
Hello to the Wikiversity community! I’m currently running for bureaucratship on the project. I’ve been part of the Wikiversity community since 2010 (at the age of 7, though not exactly sure I knew what I was doing back then…) and I’ve served as an administrator on the project since June 2021 (see my request from back then [[Wikiversity:Candidates for Custodianship/Atcovi5|here]]). I’ve also served as an English Wikibooks administrator since March 2015, a MediaWiki administrator since 2017, and held other roles previously on the Wikimedia Projects (including administrator rights on Meta Wiki and global sysopship).

I hope to continue my personal projects (see [[User:Atcovi/Works|this]] for some of these projects) and ensure that content on Wikiversity adheres to Wikiversity guidelines/policies. This includes removing/managing pseudoscientific content masquerading as established science, as well as other content that violates Wikiversity’s learning principles and guidelines. 

I'm more than happy to take up additional responsibilities to better serve the community, and I hope my past experiences in trusted positions can demonstrate my ability to handle higher responsibilities. 

Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:19, 12 May 2026 (UTC)
==== Questions ====
==== Comments ====
==== Voting ====
*{{support}} Trusted and helpful user who has shown good judgement. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:02, 12 May 2026 (UTC)
* {{support}} per the reasoning, Wikiversity could probably have more custodians and bureaucrats available. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:17, 12 May 2026 (UTC)
* {{support}} A trusted contributor to Wikiversity, custodian here for ~5 years, admin experience/roles on other wiki projects without any notable issues. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:39, 12 May 2026 (UTC)
* {{support}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:24, 13 May 2026 (UTC)
[[Category:Nominations for Bureaucratship|Atcovi]]
* {{support}} Seen your posts around, seem like you have a passion and you know what you are doing. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:43, 14 May 2026 (UTC)
*{{Oppose}} This user has been overzealous, narrow minded, and exhibited poor judgement throughout the development of Artificial Intelligent policy for Wikiversity. They have considered AI use as monolithic, failing to acknowledge and accommodate the nuances of the many ways the new technology can be used. Before the actual problem to be addressed by the policy was identified, this user defaced dozens of pages before discussing and debating policy options. More parsimonious and viable proposals were overlooked or dismissed. Requested parameterization features of the mandated macro have yet to be provided, and the present policy draws undue attention and distracts users. These are not behaviors we want to encourage within the community. --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 20:11, 15 May 2026 (UTC)
*:Context for these statements for transparency: [[Wikiversity:Colloquium/archives/January 2026#h-Template:AI-generated-20260126155300|Wikiversity:Colloquium/archives/January 2026#h-Template:AI-generated-20260126155300]], [[Wikiversity:Colloquium/archives/March 2026#h-Wikiversity:Artificial intelligence to become an official policy-20260310145400|Wikiversity:Colloquium/archives/March 2026#h-Wikiversity:Artificial intelligence to become an official policy-20260310145400]] and concerns that encouraged me to look into the matter [AI-generated content on Wikiversity] deeper include [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Inner_Development_Goals this], [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Multipolar_trap this], and [[Talk:Reformation Workshop|this]]. If there are any other discussions that I may be missing, please feel free to link them here. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:40, 15 May 2026 (UTC)
* {{support}} based on my experience with them though I don't contribute much here so feel free to discard if it doesn't count. [[User:Leaderboard|Leaderboard]] ([[User talk:Leaderboard|discuss]] • [[Special:Contributions/Leaderboard|contribs]]) 10:15, 21 May 2026 (UTC)</text>
      <sha1>el3hqslkee9b79bxdn6hnpngzl4rg2j</sha1>
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  </page>
  <page>
    <title>Wikiversity:Candidates for Custodianship/PieWriter</title>
    <ns>4</ns>
    <id>329602</id>
    <revision>
      <id>2810769</id>
      <parentid>2810526</parentid>
      <timestamp>2026-05-21T10:50:37Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Custodians offering mentorship */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2810769</origin>
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      <text bytes="2865" sha1="fu84pm16ubnbwlaqul27w2md9xdabag" xml:space="preserve">=== {{User|PieWriter}} ===
Hello everyone! I am submitting my request for temporary custodianship for 4 weeks following the recent notice seeking additional custodians. I believe I can contribute positively to the project.

If granted the tools, I would be able to block LTAs and spambots, who may try to disrupt the project. I already have curator rights, but they do not allow me to &lt;code&gt;block&lt;/code&gt; users. I would also be able to &lt;code&gt;undelete&lt;/code&gt; pages in order to help with the undeletion requests.

I already have experience working with advanced permissions on Wikiquote, where I am an administrator, so I am familiar with the responsibilities and expectations that come with administrator access. I understand the importance of using the tools carefully, and only when necessary.

Thanks for considering me :)

==== Custodians offering mentorship ====
* Hopefully someone else might step in here and mentor, but I am available. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
*:Sorry I skimmed passed this. I'm willing to mentor as I was willing to mentor beforehand in PieWriter's request for curatorship. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:39, 20 May 2026 (UTC)
* Fantastic, thankyou. Could you also list yourself here: [[Wikiversity:List of custodian mentors]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:50, 21 May 2026 (UTC)

==== Questions ====

==== Comments ====
* {{support}} PieWriter seems to know their way around wiki admin, has been contributing positively in this respect to Wikiversity, and is communicative. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
* {{comment}} See also: [[Wikiversity:Candidates for Curatorship/PieWriter]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 14 May 2026 (UTC)
* {{support}} Seems safe enough for temporary adminship/custodianship and if it's successful and PW is motivated, I would encourage indefinite user rights. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 14 May 2026 (UTC)
* {{support}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:53, 18 May 2026 (UTC)
* {{support}} satisfactory work. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:40, 20 May 2026 (UTC)</text>
      <sha1>fu84pm16ubnbwlaqul27w2md9xdabag</sha1>
    </revision>
    <revision>
      <id>2810770</id>
      <parentid>2810769</parentid>
      <timestamp>2026-05-21T10:50:55Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Custodians offering mentorship */</comment>
      <origin>2810770</origin>
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      <text bytes="2867" sha1="n3nl2fj8o7754hnuu0yd4t0lz5twt69" xml:space="preserve">=== {{User|PieWriter}} ===
Hello everyone! I am submitting my request for temporary custodianship for 4 weeks following the recent notice seeking additional custodians. I believe I can contribute positively to the project.

If granted the tools, I would be able to block LTAs and spambots, who may try to disrupt the project. I already have curator rights, but they do not allow me to &lt;code&gt;block&lt;/code&gt; users. I would also be able to &lt;code&gt;undelete&lt;/code&gt; pages in order to help with the undeletion requests.

I already have experience working with advanced permissions on Wikiquote, where I am an administrator, so I am familiar with the responsibilities and expectations that come with administrator access. I understand the importance of using the tools carefully, and only when necessary.

Thanks for considering me :)

==== Custodians offering mentorship ====
* Hopefully someone else might step in here and mentor, but I am available. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
*:Sorry I skimmed passed this. I'm willing to mentor as I was willing to mentor beforehand in PieWriter's request for curatorship. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:39, 20 May 2026 (UTC)
*:: Fantastic, thankyou. Could you also list yourself here: [[Wikiversity:List of custodian mentors]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:50, 21 May 2026 (UTC)

==== Questions ====

==== Comments ====
* {{support}} PieWriter seems to know their way around wiki admin, has been contributing positively in this respect to Wikiversity, and is communicative. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
* {{comment}} See also: [[Wikiversity:Candidates for Curatorship/PieWriter]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 14 May 2026 (UTC)
* {{support}} Seems safe enough for temporary adminship/custodianship and if it's successful and PW is motivated, I would encourage indefinite user rights. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 14 May 2026 (UTC)
* {{support}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:53, 18 May 2026 (UTC)
* {{support}} satisfactory work. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:40, 20 May 2026 (UTC)</text>
      <sha1>n3nl2fj8o7754hnuu0yd4t0lz5twt69</sha1>
    </revision>
    <revision>
      <id>2810774</id>
      <parentid>2810770</parentid>
      <timestamp>2026-05-21T11:33:49Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Custodians offering mentorship */ Reply</comment>
      <origin>2810774</origin>
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      <text bytes="3003" sha1="kpuxjqn1g5vf41vs7qi1uobqwp3l6rj" xml:space="preserve">=== {{User|PieWriter}} ===
Hello everyone! I am submitting my request for temporary custodianship for 4 weeks following the recent notice seeking additional custodians. I believe I can contribute positively to the project.

If granted the tools, I would be able to block LTAs and spambots, who may try to disrupt the project. I already have curator rights, but they do not allow me to &lt;code&gt;block&lt;/code&gt; users. I would also be able to &lt;code&gt;undelete&lt;/code&gt; pages in order to help with the undeletion requests.

I already have experience working with advanced permissions on Wikiquote, where I am an administrator, so I am familiar with the responsibilities and expectations that come with administrator access. I understand the importance of using the tools carefully, and only when necessary.

Thanks for considering me :)

==== Custodians offering mentorship ====
* Hopefully someone else might step in here and mentor, but I am available. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
*:Sorry I skimmed passed this. I'm willing to mentor as I was willing to mentor beforehand in PieWriter's request for curatorship. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:39, 20 May 2026 (UTC)
*:: Fantastic, thankyou. Could you also list yourself here: [[Wikiversity:List of custodian mentors]]? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:50, 21 May 2026 (UTC)
*:::{{done}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 11:33, 21 May 2026 (UTC)

==== Questions ====

==== Comments ====
* {{support}} PieWriter seems to know their way around wiki admin, has been contributing positively in this respect to Wikiversity, and is communicative. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:05, 14 May 2026 (UTC)
* {{comment}} See also: [[Wikiversity:Candidates for Curatorship/PieWriter]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:12, 14 May 2026 (UTC)
* {{support}} Seems safe enough for temporary adminship/custodianship and if it's successful and PW is motivated, I would encourage indefinite user rights. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 14 May 2026 (UTC)
* {{support}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 02:53, 18 May 2026 (UTC)
* {{support}} satisfactory work. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:40, 20 May 2026 (UTC)</text>
      <sha1>kpuxjqn1g5vf41vs7qi1uobqwp3l6rj</sha1>
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  <page>
    <title>User:Emmalinda1983</title>
    <ns>2</ns>
    <id>329737</id>
    <revision>
      <id>2810590</id>
      <parentid>2810423</parentid>
      <timestamp>2026-05-20T12:52:06Z</timestamp>
      <contributor>
        <username>Emmalinda1983</username>
        <id>3077930</id>
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      <comment>Update the content with more information</comment>
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      <text bytes="2371" sha1="4tseppao0h7nhlzn6ghgofj8907g832" xml:space="preserve">'''&lt;big&gt;Medical Billing Software&lt;/big&gt;'''

'''Introduction:'''

Medical billing is the process of submitting healthcare claims to insurance companies and following up on payments for medical services provided by hospitals, clinics, and healthcare professionals.

It plays an important role in the healthcare revenue cycle by helping providers receive reimbursement for patient care services. Medical billing involves insurance verification, medical coding, claim submission, payment posting, and denial management.

Medical billing is widely used by hospitals, clinics, physicians, and healthcare organizations.

'''How Medical Billing Works'''

The medical billing process usually includes the following steps:

&lt;nowiki&gt;#&lt;/nowiki&gt; Patient registration

&lt;nowiki&gt;#&lt;/nowiki&gt; Insurance verification

&lt;nowiki&gt;#&lt;/nowiki&gt; Medical coding

&lt;nowiki&gt;#&lt;/nowiki&gt; Claim submission

&lt;nowiki&gt;#&lt;/nowiki&gt; Insurance claim review

&lt;nowiki&gt;#&lt;/nowiki&gt; Payment processing

&lt;nowiki&gt;#&lt;/nowiki&gt; Denial management

&lt;nowiki&gt;#&lt;/nowiki&gt; Patient billing

&lt;big&gt;'''Key Features:'''&lt;/big&gt;

&lt;nowiki&gt;*&lt;/nowiki&gt; Insurance claim management

&lt;nowiki&gt;*&lt;/nowiki&gt; Payment tracking

&lt;nowiki&gt;*&lt;/nowiki&gt; Patient billing

&lt;nowiki&gt;*&lt;/nowiki&gt; Revenue cycle management

&lt;nowiki&gt;*&lt;/nowiki&gt; Reporting and analytics

&lt;nowiki&gt;*&lt;/nowiki&gt; Electronic health record integration

&lt;big&gt;'''Benefits:'''&lt;/big&gt; Medical billing software can help healthcare organizations:

&lt;nowiki&gt;*&lt;/nowiki&gt; Reduce billing errors

&lt;nowiki&gt;*&lt;/nowiki&gt; Improve operational efficiency

&lt;nowiki&gt;*&lt;/nowiki&gt; Speed up claim processing

&lt;nowiki&gt;*&lt;/nowiki&gt; Improve payment collection

&lt;nowiki&gt;*&lt;/nowiki&gt; Reduce paperwork

== Difference Between Medical Billing and Normal Billing ==
{| class=&quot;wikitable&quot;
!Feature
!Medical Billing
!Normal Billing
|-
|Industry
|Healthcare
|Retail and General Business
|-
|Insurance involvement
|Yes
|Usually No
|-
|Coding systems
|ICD, CPT, HCPCS codes required
|Not required
|-
|Compliance rules
|HIPAA regulations apply
|General tax regulations
|-
|Claim denials
|Common
|Rare
|-
|Processing time
|Longer
|Faster
|}
'''&lt;big&gt;Conclusion:&lt;/big&gt;'''

Medical billing software plays an important role in modern healthcare administration by improving billing accuracy, operational efficiency, and financial management.

== References ==
&lt;references /&gt;
&lt;ref&gt;https://www.cms.gov/&lt;/ref&gt;

Medical Coding

&lt;ref&gt;https://www.aapc.com/&lt;/ref&gt;</text>
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== Licensing ==
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{{Information
|Description=Laurent.5: Permutation 6C (2026520 - 20260519)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  <page>
    <title>Ship strength</title>
    <ns>0</ns>
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    <redirect title="Naval architecture/Ship strength" />
    <revision>
      <id>2810608</id>
      <timestamp>2026-05-20T14:20:22Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>Atcovi moved page [[Ship strength]] to [[Naval architecture/Ship strength]]: moving under project</comment>
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      <sha1>08aogpacneldy8h0e6hsn88pjqykexg</sha1>
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      <timestamp>2026-05-20T14:20:31Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>Atcovi moved page [[Ship Construction]] to [[Naval architecture/Ship Construction]]: moving under project</comment>
      <origin>2810610</origin>
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      <text bytes="50" sha1="go5ll4ixcj8c44ydbutzvhnnd73yqjc" xml:space="preserve">#REDIRECT [[Naval architecture/Ship Construction]]</text>
      <sha1>go5ll4ixcj8c44ydbutzvhnnd73yqjc</sha1>
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      <comment>{{Information
|Description=2. Newton-Raphson Method (20260518 - 20260512)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
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{{Information
|Description=2. Newton-Raphson Method (20260518 - 20260512)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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      <comment>{{Information
|Description=2. Newton-Raphson Method (20260519 - 20260518)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
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{{Information
|Description=2. Newton-Raphson Method (20260519 - 20260518)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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      <comment>{{Information
|Description=2. Newton-Raphson Method (20260520 - 20260519)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
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{{Information
|Description=2. Newton-Raphson Method (20260520 - 20260519)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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      <comment>{{Information
|Description=LCal.9A: Recursion (20260520 - 20260519)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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      <text bytes="262" sha1="rwxyflfo8eshgivq2137hu2zuevnti5" xml:space="preserve">== Summary ==
{{Information
|Description=LCal.9A: Recursion (20260520 - 20260519)
|Source={{own|Young1lim}}
|Date=2026-05-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}
== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  <page>
    <title>Introduction to solar energy/Solar resources and availability</title>
    <ns>0</ns>
    <id>329774</id>
    <revision>
      <id>2810664</id>
      <timestamp>2026-05-20T22:24:58Z</timestamp>
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      <comment>Created page with &quot;On average the solar radiation arriving at the top of the Earth's atmosphere is approximately 1361 W/m². No matter what time of day or year it is, the Earth, seen from the Sun as a circular disk, will receive approximately 1361 W/m² at all times.&quot;</comment>
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      <text bytes="248" sha1="pme8q51e9rd2k5fz1903irlqw0hc75f" xml:space="preserve">On average the solar radiation arriving at the top of the Earth's atmosphere is approximately 1361 W/m². No matter what time of day or year it is, the Earth, seen from the Sun as a circular disk, will receive approximately 1361 W/m² at all times.</text>
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    <revision>
      <id>2810665</id>
      <parentid>2810664</parentid>
      <timestamp>2026-05-20T22:32:59Z</timestamp>
      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
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      <text bytes="2201" sha1="h15zcuy289artgf8i5y03hc1hky5f4c" xml:space="preserve">On average the solar radiation arriving at the top of the Earth's atmosphere is approximately 1361 W/m². No matter what time of day or year it is, the Earth, seen from the Sun as a circular disk, will receive approximately 1361 W/m² at all times. For a specific location on the Earth's surface, the amount of solar irradiance received, will depend on many factors, some of which are periodic, and others, variable. 

The periodic variables, are those that repeat cyclically with periodicity measured in hours, i.e. the day, or in days, i.e. the month or the year, and others in much larger time scales, as the orbit of the Earth around itself slowly changes and as the orbit of the Earth around the Sun slowly changes.

Relevant periodic timescales for solar engineering are:

* Day/night cycles
* Seasonal cycles

Other variable, and difficult to predict factors are:

* Cloud coverage
* Other natural weather events

Luckily, there is a plethora of equations designed to predict with a high-level of accuracy, the daily paths of the sun over the horizon.

== Latitude and longitude ==
In addition to the position of the planet around the sun, the amount of sunlight available at any point on the Earth's surface will also depend on the longitude and latitude of the geographic position. Latitude is a measure of how far north or south you are, and varies from -'''90° (90° south) to +90° (90° north).''' The longitude is a measure of how far east or west one is, and varies from -180'''°''' '''(180° west) to +180° (180° east).''' Thus, at the north pole, the latitude is 90°, at the equator, the latitude is 0°, and at the south pole, the latitude is exactly -90°. 

== True solar time ==
True solar time (TST) is used to convey the relative position of the Sun compared to the Earth, irrespective of the position on Earth. Similar to the 24:00 clock, TST is measured in hours. However, specific times, such as noon, also known as, solar noon, conveys a specific and precise definition of the position of the Sun in the sky. At solar noon, the sun is at its highest point in the sky. The following equation returns the TST in hours:

&lt;math&gt;TST = UTC + \frac{longitude}{15} + EoT&lt;/math&gt;</text>
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    <revision>
      <id>2810666</id>
      <parentid>2810665</parentid>
      <timestamp>2026-05-20T22:34:52Z</timestamp>
      <contributor>
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        <id>2918363</id>
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      <comment>/* True solar time */</comment>
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      <text bytes="2392" sha1="p1ukz1uox09045rc6yct4rvfepdovhf" xml:space="preserve">On average the solar radiation arriving at the top of the Earth's atmosphere is approximately 1361 W/m². No matter what time of day or year it is, the Earth, seen from the Sun as a circular disk, will receive approximately 1361 W/m² at all times. For a specific location on the Earth's surface, the amount of solar irradiance received, will depend on many factors, some of which are periodic, and others, variable. 

The periodic variables, are those that repeat cyclically with periodicity measured in hours, i.e. the day, or in days, i.e. the month or the year, and others in much larger time scales, as the orbit of the Earth around itself slowly changes and as the orbit of the Earth around the Sun slowly changes.

Relevant periodic timescales for solar engineering are:

* Day/night cycles
* Seasonal cycles

Other variable, and difficult to predict factors are:

* Cloud coverage
* Other natural weather events

Luckily, there is a plethora of equations designed to predict with a high-level of accuracy, the daily paths of the sun over the horizon.

== Latitude and longitude ==
In addition to the position of the planet around the sun, the amount of sunlight available at any point on the Earth's surface will also depend on the longitude and latitude of the geographic position. Latitude is a measure of how far north or south you are, and varies from -'''90° (90° south) to +90° (90° north).''' The longitude is a measure of how far east or west one is, and varies from -180'''°''' '''(180° west) to +180° (180° east).''' Thus, at the north pole, the latitude is 90°, at the equator, the latitude is 0°, and at the south pole, the latitude is exactly -90°. 

== True solar time ==
True solar time (TST) is used to convey the relative position of the Sun compared to the Earth, irrespective of the position on Earth. Similar to the 24:00 clock, TST is measured in hours. However, specific times, such as noon, also known as, solar noon, conveys a specific and precise definition of the position of the Sun in the sky. At solar noon, the sun is at its highest point in the sky. The following equation returns the TST in hours:

&lt;math&gt;TST = UTC + \frac{longitude}{15} + EoT&lt;/math&gt;

Where UTC stands for Universal Time Coordinates, longitude is measured in degrees, and EoT represents the equation of time in minutes, defined later in this chapter.

== Equation of time ==</text>
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  <page>
    <title>Talk:Introduction to solar energy/Solar resources and availability</title>
    <ns>1</ns>
    <id>329775</id>
    <revision>
      <id>2810667</id>
      <timestamp>2026-05-20T22:35:37Z</timestamp>
      <contributor>
        <username>IanVG</username>
        <id>2918363</id>
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      <comment>/* Moving and improving content */ new section</comment>
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      <text bytes="291" sha1="gi9x7g50b68dsethxupts44uy8zvtl7" xml:space="preserve">== Moving and improving content ==

I moved (copied and improved) content from [[Solar energy/Resource]], to this page as part of the introduction to solar energy course. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:35, 20 May 2026 (UTC)</text>
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      <id>2810668</id>
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      <timestamp>2026-05-20T22:35:59Z</timestamp>
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      <minor />
      <comment>/* Moving and improving content */</comment>
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      <text bytes="304" sha1="e0gled2qen7fml1aamc6jttab3mk8kj" xml:space="preserve">== Moving and improving content ==

I moved (copied and improved) content from [[Solar energy/Resource availability]], to this page as part of the introduction to solar energy course. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:35, 20 May 2026 (UTC)</text>
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  <page>
    <title>Talk:Solar energy/Resource availability</title>
    <ns>1</ns>
    <id>329776</id>
    <revision>
      <id>2810669</id>
      <timestamp>2026-05-20T22:37:18Z</timestamp>
      <contributor>
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        <id>2918363</id>
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      <comment>/* Moved and improved content */ new section</comment>
      <origin>2810669</origin>
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      <text bytes="350" sha1="jx3e341px3qtpqaih79ygby1eqpcpf1" xml:space="preserve">== Moved and improved content ==

I started moving and improving content from this page to [[Introduction to solar energy/Solar resources and availability]] in an effort to better flesh out the introductory course I'm working on. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:37, 20 May 2026 (UTC)</text>
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  <page>
    <title>Introduction to solar energy/Equations</title>
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      <id>2810672</id>
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      <comment>Created page with &quot;== Solar resources and availability == &lt;math&gt;TST = UTC + \frac{longitude}{15} + EoT&lt;/math&gt;  === Precise method === &lt;math&gt;EoT = \Delta t_{ey} =  -7.659\sin(D) + 9.863\sin \left(2D + 3.5932 \right)&lt;/math&gt;(minutes)  where: &lt;math&gt;D = 6.240\, 040\, 77 + 0.017\, 201\, 97(365.25(y-2000) + d)&lt;/math&gt;  where &lt;math&gt;d&lt;/math&gt; represents the number of days since 1 January of the current year, &lt;math&gt;y&lt;/math&gt;.  === Rough approximation === For &lt;math&gt;EoT&lt;/math&gt; in minutes, see the followi...&quot;</comment>
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      <text bytes="2697" sha1="db6q9kukv3nyq0pl6j28lqcmzvg0f9o" xml:space="preserve">== Solar resources and availability ==
&lt;math&gt;TST = UTC + \frac{longitude}{15} + EoT&lt;/math&gt;

=== Precise method ===
&lt;math&gt;EoT = \Delta t_{ey} =  -7.659\sin(D) + 9.863\sin \left(2D + 3.5932 \right)&lt;/math&gt;(minutes)

where: &lt;math&gt;D = 6.240\, 040\, 77 + 0.017\, 201\, 97(365.25(y-2000) + d)&lt;/math&gt;

where &lt;math&gt;d&lt;/math&gt; represents the number of days since 1 January of the current year, &lt;math&gt;y&lt;/math&gt;.

=== Rough approximation ===
For &lt;math&gt;EoT&lt;/math&gt; in minutes, see the following equation. Ensure that the trigonometry relations are calculated in ''radians'' mode.

&lt;math&gt;EoT (minutes) = \Delta t_{ey} = 7.678\sin ({B + 1.374}) - 9.87\sin ({2B})&lt;/math&gt;

Where B is:

&lt;math&gt;B = 2 \pi \frac {(n-81)}{365}&lt;/math&gt;

Where &lt;math&gt;n&lt;/math&gt; is the day of the year.

For &lt;math&gt;EoT&lt;/math&gt; in degrees, see the following equation. Ensure that the trigonometry relations are calculated in ''degrees'' mode.

&lt;math&gt;EoT (degrees) = 0.0072cos(J) - 0.0528cos(2J) -0.0012cos(3J) - 0.1229sin(J) - 0.1565sin(2J) - 0.0041sin(3J)&lt;/math&gt;Where J is:

&lt;math&gt;J = n \times \frac {360}{365}&lt;/math&gt;

Where &lt;math&gt;n&lt;/math&gt; is the day of the year.

=== Solar declination rough approximation, in degrees ===

=== &lt;math&gt;\delta_{degrees}= 23.45sin(360(n+284)/365)&lt;/math&gt; ===

=== Solar declination finer result, in radians ===
&lt;math&gt;\delta_{radians}= arcsin(0.369sin(360(n-82)/365 +2sin(360(n-2)/365))&lt;/math&gt;

=== Solar declination finer result, in degrees ===
&lt;math&gt;\delta_{radians}= 0.33281 - 22.984cos(J) - 0.3499 cos(2J) - 0.1398cos(3J) + 3.7872sin(J) + 0.03205sin(2J) + 0.07187(3J0&lt;/math&gt;

Where J is the day of the year, with January 1st being n=1.

&lt;math&gt;J = n \times (360/365)&lt;/math&gt;, in degrees.

=== Half hour angle, ω₀ ===
Calculate the hour angle when the altitude angle is equal to 0.

&lt;math&gt;\omega_0 (\text {degrees})= arccos(-(\tan {\delta})* (\tan {Lat}))&lt;/math&gt;

Using this equation, you can find true solar time (TST) at sunrise and at sunset.

&lt;math&gt;TST_{sunrise} = 12 - (\frac {1}{15}) \omega_0&lt;/math&gt;, make sure that &lt;math&gt;\omega_0&lt;/math&gt;is in degrees.

&lt;math&gt;TST_{sunset} = 12 + (\frac {1}{15}) \omega_0&lt;/math&gt;, make sure that &lt;math&gt;\omega_0&lt;/math&gt;is in degrees.

&lt;math&gt;S_0 = (\frac {2}{15}) \omega_0&lt;/math&gt;, make sure that &lt;math&gt;\omega_0&lt;/math&gt;is in degrees.

if: &lt;math&gt;\omega_0 &gt; 90&lt;/math&gt;, then the day is longer than the night, and the time that the sun is in the sky is at least 12 hours.

if: &lt;math&gt;\omega_0 &lt; 90&lt;/math&gt;, then the day is shorter than the night, and the time that the sun is in the sky is less than 12 hours.

=== Elevation angle, h ===
&lt;math&gt;h = arcsin[cos(\delta - {Lat})] = 90^{\circ} - Lat + \delta&lt;/math&gt;

&lt;math&gt;h = arcsin(\cos {\delta}\cos{HA}\cos{Lat} +\sin{\delta}\sin{Lat})&lt;/math&gt;</text>
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  <page>
    <title>United States UFO files</title>
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      <comment>Page created</comment>
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      <text bytes="8057" sha1="a2d0vi7d895q4b91yqctnffs66kyzog" xml:space="preserve">[[File:UAP Photo December 2025.jpg|thumb|A UFO over western United States in December, 2025]]
The '''United States UFO files''' are a collection of declassified United States government records concerning UFOs, also called unidentified anomalous phenomena (UAPs), released by the administration of [[President of the United States/Donald Trump|Donald Trump]] beginning on May 8, 2026, with plans for ongoing and expanding releases of additional materials.

Belief that the U.S. Government is concealing information related to non-human intelligence, aliens or extraterrestrials visiting Earth is a long-running conspiracy theory.&lt;ref&gt;{{cite web|url=https://www.ebsco.com/research-starters/visual-arts/ufo-evidence-and-censorship|title=UFO Evidence and Censorship|website=Research Starters|publisher=[[EBSCO Information Services|EBSCO]]|archive-url=https://web.archive.org/web/20260326223452/https://www.ebsco.com/research-starters/visual-arts/ufo-evidence-and-censorship|archive-date=March 26, 2026|access-date=May 8, 2026|url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|last1=Canavan|first1=Gerru|date=2025|title=The Truth Is Out There: UFO Conspiracy and Science Fiction|url=https://epublications.marquette.edu/english_fac/640/|journal=Journal of Cinema and Media Studies|volume=64|issue=3|pages=150-155|access-date=May 8, 2026}}&lt;/ref&gt;

Beginning in 2017, the ''[[wikipedia:New_York_Times|New York Times]]'' and other outlets disclosed the Pentagon UFO videos and secret government programs to investigate &quot;unusual, seemingly inexplicable phenomena&quot;, as ''The Washington Post'' described them.&lt;ref name=&quot;WAPO UFO 2026-05-08&quot;&gt;{{Cite news|url=https://www.washingtonpost.com/national-security/2026/05/08/trump-ufo-files-release/|title=Trump releases UFO files, says public can judge for themselves|last=Diamond|first=Dan|date=2026-05-08|work=The Washington Post|access-date=2026-05-08|archive-url=https://web.archive.org/web/20260511215734/https://www.washingtonpost.com/national-security/2026/05/08/trump-ufo-files-release/|archive-date=2026-05-11|language=en-US|issn=0190-8286|quote=The release also follows years of news reports and government releases that have shifted officials' willingness to openly engage with the possibility of extraterrestrial life. Beginning in 2017, the New York Times and other media outlets have run reports on secret federal programs that have studied unusual, seemingly inexplicable phenomena. Government agencies have released videos of aircraft that appeared to defy the laws of physics. Even past presidents have acknowledged their questions about potential government cover-ups.|url-access=subscription}}&lt;/ref&gt; Bipartisan pressure in Congress combined with military interests around unexplained UFO encounters for multiple years had increased related to reports of incidents by defense and intelligence officials, radar operators, and pilots.&lt;ref name=&quot;Military UFOs 2026-05-08&quot;&gt;{{Cite web|url=https://www.military.com/trump-opens-ufo-files-in-historic-government-release|title=Trump Opens UFO Files in Historic Government Release|last=Radzius|first=Darius|date=2026-05-08|website=Military.com|language=en|archive-url=https://web.archive.org/web/20260509030757/https://www.military.com/trump-opens-ufo-files-in-historic-government-release|archive-date=2026-05-09|access-date=2026-05-08}}&lt;/ref&gt; Congress had previously created the All-domain Anomaly Resolution Office in 2022 to &quot;investigate anomalous incidents observed across air, maritime, space and other domains&quot;, reported Military.com.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;

In 2022, [[wikipedia:NASA|NASA]] conducted an inquiry into the potential of space aliens visiting Earth and found no evidence to support such a notion.&lt;ref&gt;{{cite news|url=https://www.livescience.com/space/extraterrestrial-life/us-government-declassifies-nearly-200-uap-files-including-strange-sightings-from-apollo-astronauts|title=US government declassifies nearly 200 UAP files, including strange sightings from Apollo astronauts|last1=Specktor|first1=Brandon|date=May 8, 2026|work=[[Live Science]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260509012940/https://www.livescience.com/space/extraterrestrial-life/us-government-declassifies-nearly-200-uap-files-including-strange-sightings-from-apollo-astronauts|archive-date=May 9, 2026|url-status=live}}&lt;/ref&gt; Two years later, in 2024, the U.S. Department of Defense officially disclaimed it had any information pertaining to the existence of extraterrestrial life or that it had recovered technology belonging to space aliens.&lt;ref name=&quot;apn&quot;&gt;{{cite news|url=https://apnews.com/article/trump-ufos-uap-aliens-pentagon-records-investigation-3e658d2cf3742465127c0049c872240a|title=Bright lights and hot orbs: UFO files shed light on sightings but leave interpretation to the public|last1=Binkley|first1=Collin|date=May 8, 2026|work=[[Associated Press]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260508130822/https://apnews.com/article/trump-ufos-uap-aliens-pentagon-records-investigation-3e658d2cf3742465127c0049c872240a|archive-date=May 8, 2026|url-status=live}}&lt;/ref&gt; Donald Trump, Barack Obama, and Bill Clinton have all stated their disbelief in any anomalous explanation for UFOs and denied the existence of any U.S. government secrecy surrounding the topic.&lt;ref&gt;{{cite news|url=https://abcnews.com/Politics/exclusive-trump-unidentified-flying-objects/story?id=63725191|title=EXCLUSIVE: Trump says he doesn't particularly believe in unidentified flying objects|last1=McLaughlin|first1=Elizabeth|work=[[ABC News (United States)|ABC News]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260329233503/https://abcnews.com/Politics/exclusive-trump-unidentified-flying-objects/story?id=63725191|archive-date=March 29, 2026|volume=June 15, 2019|url-status=live}}&lt;/ref&gt;&lt;ref name=&quot;nytimes1&quot;&gt;{{cite news|url=https://www.nytimes.com/2026/05/08/us/politics/pentagon-ufo-files.html|title=Pentagon Releases Files on U.F.O.s|last1=Cooper|first1=Helene|date=May 8, 2025|work=[[New York Times]]|access-date=May 8, 2025|author-link=Helene Cooper}}&lt;/ref&gt; U.S. government agencies have released videos of aircraft, which, according to ''The Washington Post'', &quot;appeared to defy the laws of physics&quot;.&lt;ref name=&quot;WAPO UFO 2026-05-08&quot; /&gt; Congressional hearings in the past decade included testimony from government, military, and intelligence witness of UFO events, while skeptics claimed the U.S. military and intelligence reports and data were unreliable.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;

In February 2026, U.S. President Donald Trump ordered federal agencies to identify and declassify to the public files connected to UFOs, UAPs, and extraterrestrials.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;&lt;ref name=&quot;WaPo080526&quot;&gt;{{cite news|url=https://www.washingtonpost.com/politics/2026/02/19/trump-obama-ufo-classified/|title=Trump says he will release government files on alien life, UFOs|last1=Diamond|first1=Dan|date=February 19, 2026|work=[[Washington Post]]|access-date=May 8, 2026}}&lt;/ref&gt; The move followed agitation by several Republican members of the U.S. Congress, such as Anna Paulina Luna and Tim Burchett, who claimed the U.S. government was withholding secrets about space aliens.&lt;ref name=&quot;ap&quot;&gt;{{cite news|url=https://www.pbs.org/newshour/politics/trump-drops-hints-of-whats-coming-in-new-batch-of-ufo-files-set-for-release|title=Trump drops hints of what's coming in new batch of UFO files set for release|date=May 4, 2026|work=[[PBS]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260508065205/https://www.pbs.org/newshour/politics/trump-drops-hints-of-whats-coming-in-new-batch-of-ufo-files-set-for-release|archive-date=May 8, 2026|agency=[[Associated Press]]|url-status=live}}&lt;/ref&gt;&lt;ref name=&quot;apn&quot; /&gt; The month following Trump's directive, Luna demanded Secretary of Defense Pete Hegseth send her some videos of UFOs by a deadline she established; according to the Associated Press, Luna's demand &quot;came and went, and no videos were produced&quot;.&lt;ref name=&quot;ap&quot; /&gt;</text>
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[[File:UAP Photo December 2025.jpg|thumb|A UFO over western United States in December, 2025]]
The '''United States UFO files''' are a collection of declassified United States government records concerning UFOs, also called unidentified anomalous phenomena (UAPs), released by the administration of [[President of the United States/Donald Trump|Donald Trump]] beginning on May 8, 2026, with plans for ongoing and expanding releases of additional materials.

Belief that the U.S. Government is concealing information related to non-human intelligence, aliens or extraterrestrials visiting Earth is a long-running conspiracy theory.&lt;ref&gt;{{cite web|url=https://www.ebsco.com/research-starters/visual-arts/ufo-evidence-and-censorship|title=UFO Evidence and Censorship|website=Research Starters|publisher=[[EBSCO Information Services|EBSCO]]|archive-url=https://web.archive.org/web/20260326223452/https://www.ebsco.com/research-starters/visual-arts/ufo-evidence-and-censorship|archive-date=March 26, 2026|access-date=May 8, 2026|url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|last1=Canavan|first1=Gerru|date=2025|title=The Truth Is Out There: UFO Conspiracy and Science Fiction|url=https://epublications.marquette.edu/english_fac/640/|journal=Journal of Cinema and Media Studies|volume=64|issue=3|pages=150-155|access-date=May 8, 2026}}&lt;/ref&gt;

Beginning in 2017, the ''[[wikipedia:New_York_Times|New York Times]]'' and other outlets disclosed the Pentagon UFO videos and secret government programs to investigate &quot;unusual, seemingly inexplicable phenomena&quot;, as ''The Washington Post'' described them.&lt;ref name=&quot;WAPO UFO 2026-05-08&quot;&gt;{{Cite news|url=https://www.washingtonpost.com/national-security/2026/05/08/trump-ufo-files-release/|title=Trump releases UFO files, says public can judge for themselves|last=Diamond|first=Dan|date=2026-05-08|work=The Washington Post|access-date=2026-05-08|archive-url=https://web.archive.org/web/20260511215734/https://www.washingtonpost.com/national-security/2026/05/08/trump-ufo-files-release/|archive-date=2026-05-11|language=en-US|issn=0190-8286|quote=The release also follows years of news reports and government releases that have shifted officials' willingness to openly engage with the possibility of extraterrestrial life. Beginning in 2017, the New York Times and other media outlets have run reports on secret federal programs that have studied unusual, seemingly inexplicable phenomena. Government agencies have released videos of aircraft that appeared to defy the laws of physics. Even past presidents have acknowledged their questions about potential government cover-ups.|url-access=subscription}}&lt;/ref&gt; Bipartisan pressure in Congress combined with military interests around unexplained UFO encounters for multiple years had increased related to reports of incidents by defense and intelligence officials, radar operators, and pilots.&lt;ref name=&quot;Military UFOs 2026-05-08&quot;&gt;{{Cite web|url=https://www.military.com/trump-opens-ufo-files-in-historic-government-release|title=Trump Opens UFO Files in Historic Government Release|last=Radzius|first=Darius|date=2026-05-08|website=Military.com|language=en|archive-url=https://web.archive.org/web/20260509030757/https://www.military.com/trump-opens-ufo-files-in-historic-government-release|archive-date=2026-05-09|access-date=2026-05-08}}&lt;/ref&gt; Congress had previously created the All-domain Anomaly Resolution Office in 2022 to &quot;investigate anomalous incidents observed across air, maritime, space and other domains&quot;, reported Military.com.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;

In 2022, [[wikipedia:NASA|NASA]] conducted an inquiry into the potential of space aliens visiting Earth and found no evidence to support such a notion.&lt;ref&gt;{{cite news|url=https://www.livescience.com/space/extraterrestrial-life/us-government-declassifies-nearly-200-uap-files-including-strange-sightings-from-apollo-astronauts|title=US government declassifies nearly 200 UAP files, including strange sightings from Apollo astronauts|last1=Specktor|first1=Brandon|date=May 8, 2026|work=[[Live Science]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260509012940/https://www.livescience.com/space/extraterrestrial-life/us-government-declassifies-nearly-200-uap-files-including-strange-sightings-from-apollo-astronauts|archive-date=May 9, 2026|url-status=live}}&lt;/ref&gt; Two years later, in 2024, the U.S. Department of Defense officially disclaimed it had any information pertaining to the existence of extraterrestrial life or that it had recovered technology belonging to space aliens.&lt;ref name=&quot;apn&quot;&gt;{{cite news|url=https://apnews.com/article/trump-ufos-uap-aliens-pentagon-records-investigation-3e658d2cf3742465127c0049c872240a|title=Bright lights and hot orbs: UFO files shed light on sightings but leave interpretation to the public|last1=Binkley|first1=Collin|date=May 8, 2026|work=[[Associated Press]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260508130822/https://apnews.com/article/trump-ufos-uap-aliens-pentagon-records-investigation-3e658d2cf3742465127c0049c872240a|archive-date=May 8, 2026|url-status=live}}&lt;/ref&gt; Donald Trump, Barack Obama, and Bill Clinton have all stated their disbelief in any anomalous explanation for UFOs and denied the existence of any U.S. government secrecy surrounding the topic.&lt;ref&gt;{{cite news|url=https://abcnews.com/Politics/exclusive-trump-unidentified-flying-objects/story?id=63725191|title=EXCLUSIVE: Trump says he doesn't particularly believe in unidentified flying objects|last1=McLaughlin|first1=Elizabeth|work=[[ABC News (United States)|ABC News]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260329233503/https://abcnews.com/Politics/exclusive-trump-unidentified-flying-objects/story?id=63725191|archive-date=March 29, 2026|volume=June 15, 2019|url-status=live}}&lt;/ref&gt;&lt;ref name=&quot;nytimes1&quot;&gt;{{cite news|url=https://www.nytimes.com/2026/05/08/us/politics/pentagon-ufo-files.html|title=Pentagon Releases Files on U.F.O.s|last1=Cooper|first1=Helene|date=May 8, 2025|work=[[New York Times]]|access-date=May 8, 2025|author-link=Helene Cooper}}&lt;/ref&gt; U.S. government agencies have released videos of aircraft, which, according to ''The Washington Post'', &quot;appeared to defy the laws of physics&quot;.&lt;ref name=&quot;WAPO UFO 2026-05-08&quot; /&gt; Congressional hearings in the past decade included testimony from government, military, and intelligence witness of UFO events, while skeptics claimed the U.S. military and intelligence reports and data were unreliable.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;

In February 2026, U.S. President Donald Trump ordered federal agencies to identify and declassify to the public files connected to UFOs, UAPs, and extraterrestrials.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;&lt;ref name=&quot;WaPo080526&quot;&gt;{{cite news|url=https://www.washingtonpost.com/politics/2026/02/19/trump-obama-ufo-classified/|title=Trump says he will release government files on alien life, UFOs|last1=Diamond|first1=Dan|date=February 19, 2026|work=[[Washington Post]]|access-date=May 8, 2026}}&lt;/ref&gt; The move followed agitation by several Republican members of the U.S. Congress, such as Anna Paulina Luna and Tim Burchett, who claimed the U.S. government was withholding secrets about space aliens.&lt;ref name=&quot;ap&quot;&gt;{{cite news|url=https://www.pbs.org/newshour/politics/trump-drops-hints-of-whats-coming-in-new-batch-of-ufo-files-set-for-release|title=Trump drops hints of what's coming in new batch of UFO files set for release|date=May 4, 2026|work=[[PBS]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260508065205/https://www.pbs.org/newshour/politics/trump-drops-hints-of-whats-coming-in-new-batch-of-ufo-files-set-for-release|archive-date=May 8, 2026|agency=[[Associated Press]]|url-status=live}}&lt;/ref&gt;&lt;ref name=&quot;apn&quot; /&gt; The month following Trump's directive, Luna demanded Secretary of Defense Pete Hegseth send her some videos of UFOs by a deadline she established; according to the Associated Press, Luna's demand &quot;came and went, and no videos were produced&quot;.&lt;ref name=&quot;ap&quot; /&gt;</text>
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[[File:UAP Photo December 2025.jpg|thumb|A UFO over western United States in December, 2025]]
The '''United States UFO files''' are a collection of declassified United States government records concerning UFOs, also called unidentified anomalous phenomena (UAPs), released by the administration of [[President of the United States/Donald Trump|Donald Trump]] beginning on May 8, 2026, with plans for ongoing and expanding releases of additional materials.

Belief that the U.S. Government is concealing information related to non-human intelligence, aliens or extraterrestrials visiting Earth is a long-running conspiracy theory.&lt;ref&gt;{{cite web|url=https://www.ebsco.com/research-starters/visual-arts/ufo-evidence-and-censorship|title=UFO Evidence and Censorship|website=Research Starters|publisher=[[EBSCO Information Services|EBSCO]]|archive-url=https://web.archive.org/web/20260326223452/https://www.ebsco.com/research-starters/visual-arts/ufo-evidence-and-censorship|archive-date=March 26, 2026|access-date=May 8, 2026|url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|last1=Canavan|first1=Gerru|date=2025|title=The Truth Is Out There: UFO Conspiracy and Science Fiction|url=https://epublications.marquette.edu/english_fac/640/|journal=Journal of Cinema and Media Studies|volume=64|issue=3|pages=150-155|access-date=May 8, 2026}}&lt;/ref&gt;

Beginning in 2017, the ''[[wikipedia:New_York_Times|New York Times]]'' and other outlets disclosed the Pentagon UFO videos and secret government programs to investigate &quot;unusual, seemingly inexplicable phenomena&quot;, as ''The Washington Post'' described them.&lt;ref name=&quot;WAPO UFO 2026-05-08&quot;&gt;{{Cite news|url=https://www.washingtonpost.com/national-security/2026/05/08/trump-ufo-files-release/|title=Trump releases UFO files, says public can judge for themselves|last=Diamond|first=Dan|date=2026-05-08|work=The Washington Post|access-date=2026-05-08|archive-url=https://web.archive.org/web/20260511215734/https://www.washingtonpost.com/national-security/2026/05/08/trump-ufo-files-release/|archive-date=2026-05-11|language=en-US|issn=0190-8286|quote=The release also follows years of news reports and government releases that have shifted officials' willingness to openly engage with the possibility of extraterrestrial life. Beginning in 2017, the New York Times and other media outlets have run reports on secret federal programs that have studied unusual, seemingly inexplicable phenomena. Government agencies have released videos of aircraft that appeared to defy the laws of physics. Even past presidents have acknowledged their questions about potential government cover-ups.|url-access=subscription}}&lt;/ref&gt; Bipartisan pressure in Congress combined with military interests around unexplained UFO encounters for multiple years had increased related to reports of incidents by defense and intelligence officials, radar operators, and pilots.&lt;ref name=&quot;Military UFOs 2026-05-08&quot;&gt;{{Cite web|url=https://www.military.com/trump-opens-ufo-files-in-historic-government-release|title=Trump Opens UFO Files in Historic Government Release|last=Radzius|first=Darius|date=2026-05-08|website=Military.com|language=en|archive-url=https://web.archive.org/web/20260509030757/https://www.military.com/trump-opens-ufo-files-in-historic-government-release|archive-date=2026-05-09|access-date=2026-05-08}}&lt;/ref&gt; Congress had previously created the All-domain Anomaly Resolution Office in 2022 to &quot;investigate anomalous incidents observed across air, maritime, space and other domains&quot;, reported Military.com.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;

In 2022, [[wikipedia:NASA|NASA]] conducted an inquiry into the potential of space aliens visiting Earth and found no evidence to support such a notion.&lt;ref&gt;{{cite news|url=https://www.livescience.com/space/extraterrestrial-life/us-government-declassifies-nearly-200-uap-files-including-strange-sightings-from-apollo-astronauts|title=US government declassifies nearly 200 UAP files, including strange sightings from Apollo astronauts|last1=Specktor|first1=Brandon|date=May 8, 2026|work=[[Live Science]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260509012940/https://www.livescience.com/space/extraterrestrial-life/us-government-declassifies-nearly-200-uap-files-including-strange-sightings-from-apollo-astronauts|archive-date=May 9, 2026|url-status=live}}&lt;/ref&gt; Two years later, in 2024, the U.S. Department of Defense officially disclaimed it had any information pertaining to the existence of extraterrestrial life or that it had recovered technology belonging to space aliens.&lt;ref name=&quot;apn&quot;&gt;{{cite news|url=https://apnews.com/article/trump-ufos-uap-aliens-pentagon-records-investigation-3e658d2cf3742465127c0049c872240a|title=Bright lights and hot orbs: UFO files shed light on sightings but leave interpretation to the public|last1=Binkley|first1=Collin|date=May 8, 2026|work=[[Associated Press]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260508130822/https://apnews.com/article/trump-ufos-uap-aliens-pentagon-records-investigation-3e658d2cf3742465127c0049c872240a|archive-date=May 8, 2026|url-status=live}}&lt;/ref&gt; Donald Trump, Barack Obama, and Bill Clinton have all stated their disbelief in any anomalous explanation for UFOs and denied the existence of any U.S. government secrecy surrounding the topic.&lt;ref&gt;{{cite news|url=https://abcnews.com/Politics/exclusive-trump-unidentified-flying-objects/story?id=63725191|title=EXCLUSIVE: Trump says he doesn't particularly believe in unidentified flying objects|last1=McLaughlin|first1=Elizabeth|work=[[ABC News (United States)|ABC News]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260329233503/https://abcnews.com/Politics/exclusive-trump-unidentified-flying-objects/story?id=63725191|archive-date=March 29, 2026|volume=June 15, 2019|url-status=live}}&lt;/ref&gt;&lt;ref name=&quot;nytimes1&quot;&gt;{{cite news|url=https://www.nytimes.com/2026/05/08/us/politics/pentagon-ufo-files.html|title=Pentagon Releases Files on U.F.O.s|last1=Cooper|first1=Helene|date=May 8, 2025|work=[[New York Times]]|access-date=May 8, 2025|author-link=Helene Cooper}}&lt;/ref&gt; U.S. government agencies have released videos of aircraft, which, according to ''The Washington Post'', &quot;appeared to defy the laws of physics&quot;.&lt;ref name=&quot;WAPO UFO 2026-05-08&quot; /&gt; Congressional hearings in the past decade included testimony from government, military, and intelligence witness of UFO events, while skeptics claimed the U.S. military and intelligence reports and data were unreliable.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;

In February 2026, U.S. President Donald Trump ordered federal agencies to identify and declassify to the public files connected to UFOs, UAPs, and extraterrestrials.&lt;ref name=&quot;Military UFOs 2026-05-08&quot; /&gt;&lt;ref name=&quot;WaPo080526&quot;&gt;{{cite news|url=https://www.washingtonpost.com/politics/2026/02/19/trump-obama-ufo-classified/|title=Trump says he will release government files on alien life, UFOs|last1=Diamond|first1=Dan|date=February 19, 2026|work=[[Washington Post]]|access-date=May 8, 2026}}&lt;/ref&gt; The move followed agitation by several Republican members of the U.S. Congress, such as Anna Paulina Luna and Tim Burchett, who claimed the U.S. government was withholding secrets about space aliens.&lt;ref name=&quot;ap&quot;&gt;{{cite news|url=https://www.pbs.org/newshour/politics/trump-drops-hints-of-whats-coming-in-new-batch-of-ufo-files-set-for-release|title=Trump drops hints of what's coming in new batch of UFO files set for release|date=May 4, 2026|work=[[PBS]]|access-date=May 8, 2026|archive-url=https://web.archive.org/web/20260508065205/https://www.pbs.org/newshour/politics/trump-drops-hints-of-whats-coming-in-new-batch-of-ufo-files-set-for-release|archive-date=May 8, 2026|agency=[[Associated Press]]|url-status=live}}&lt;/ref&gt;&lt;ref name=&quot;apn&quot; /&gt; The month following Trump's directive, Luna demanded Secretary of Defense Pete Hegseth send her some videos of UFOs by a deadline she established; according to the Associated Press, Luna's demand &quot;came and went, and no videos were produced&quot;.&lt;ref name=&quot;ap&quot; /&gt;</text>
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      <comment>Added school of electronic engineering</comment>
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